Inside and Outside Liquidity∗
Patrick Bolton
Tano Santos
Jose Scheinkman
Columbia University
Columbia University
Princeton University
July 2008
Abstract
We consider a model of liquidity demand arising from maturity mismatch on one side
of the market. This demand can be met with either cash held by those with the liquidity
need, what we refer to as inside liquidity, or via asset sales for cash held by agents other
than those with liquidity needs. We refer to this cash as outside liquidity. The questions
we address are: (a) what determines the mix of inside and outside liquidity in equilibrium?
(b) does the market provide an efficient mix of inside and outside liquidity? and (c) if
not, what kind of interventions can be proposed to restore efficiency? We argue that a key
determinant of the optimal liquidity mix is the expected timing of asset sale decisions. An
important source of inefficiency in our model is the presence of asymmetric information
about asset values, which increases the longer a liquidity trade is delayed. We establish
that an immediate-trading equilibrium, in which asset trading occurs in anticipation of a
liquidity shock, always exists. Another equilibrium with delayed-trading, in which asset
trades are a response to a liquidity shock, may also exist. We show that, when it exists,
the delayed-trading equilibrium is efficient, despite the presence of adverse selection.
∗
Preliminary and incomplete. Please do not quote without permission. We thank Rafael Repullo and
Lasse Pedersen as well as participants at workshops and seminars at Columbia University, Toulouse School of
Economics and the 2008 NBER Summer Institute on Risks of Financial Institutions for their comments and
suggestions.
INTRODUCTION
The main goal of this paper is to propose a tractable model of maturity transformation by
financial intermediaries and the resulting liquidity demand arising from the maturity mismatch
between assets and liabilities. When financial intermediaries invest in long-term assets but
potentially face redemptions before these assets mature they have a need for liquidity. These
redemptions can be met either out of cash holdings of the financial intermediaries – what we
refer to as inside liquidity – or out of the proceeds from asset sales to other investors with a
longer horizon–what we refer to as outside liquidity. In reality financial intermediaries rely on
both forms of liquidity and the purpose of our analysis is to determine the relative importance
and efficiency of inside and outside liquidity in a competitive equilibrium of the financial sector.
Our model comprises two different groups of agents that differ in their investment horizons. One class of agents, which we denominate short run investors, prefer early to late payoffs,
whereas the second class, which we refer to as long run investors, are indifferent as to the timing
of the payoffs associated with their investments. One can think of these long run investors as
wealthy individuals, endowments, hedge funds and even sovereign wealth funds and the short
run investors as financial intermediaries with short dated liabilities. Short run investors allocate their investments between long-term and liquid assets, or cash, which they carry in case
their long term investments do not pay in time. If they do not, short run investors can supply
some or all of the long term assets in their books in exchange for cash. We refer to the cash
carried by the short run investors as inside liquidity. Long-horizon investors directly invest
in a portfolio of long-term and liquid assets of their own, which is also cash. These investors
carry cash, which we refer to as outside liquidity, precisely to opportunistically acquire the
long assets of the short run investors at low prices.
Within this model the key questions we are interested in are: first, what determines the
mix of inside and outside liquidity in equilibrium? second, does the market provide an efficient
mix of inside versus inside liquidity? and, third, if not, what kind of interventions can be
proposed to restore efficiency?
Our model attempts to describe situations in which short run investors hold relatively
sophisticated assets or securities, and where long run investors have sufficient expertise with
these securities to stand ready to buy them at a relatively good price. Other investors may be
ready to buy these securities, but at a much higher discount.
Still, an important potential source of inefficiency in practice and in our model remains
asymmetric information about asset values between short and long horizon investors. That is,
even when short run investors turn to experienced long run ones to sell claims to their assets,
1
the latter cannot always tell whether the sale is due to a sudden liquidity need or whether
the financial intermediary is trying to pass on a lemon. This problem is familiar to financial
market participants and has been widely studied in the literature in different contexts. The
novel aspect our model focuses on is a timing dimension. Short run investors learn more about
the underlying value of their assets over time. Therefore, when at the onset of a liquidity
shock they choose to hold on to their positions – in the hope of riding out a temporary crisis
– they run the risk of having to go to the market in a much worse position should the crisis
be a prolonged one. The longer they wait the worse is the lemons problem and therefore the
greater is the risk that they will have to sell assets at fire-sale prices.
This is a common dynamic in liquidity crises, which has not been much analyzed nor
previously modeled, and which is a core mechanism in our analysis. We capture the essence
of this dynamic unfolding of a liquidity crisis by establishing the existence of two types of
rational expectations equilibria: an immediate trading equilibrium, where short run investors
are rationally expected to trade at the onset of the liquidity shock and a delayed trading
equilibrium, where they are instead correctly expected to prefer attempting to ride out the
crisis and to only trade as a last resort should the crisis be a prolonged one. We show that for
some parameter values only the immediate trading equilibrium exists, while for other values
both equilibria coexist.
When two different rational expectations equilibria can coexist one naturally wonders
how they compare in terms of efficiency. Which is better? Interestingly, the answer to this
question turns critically on the implications for ex-ante portfolio-composition decisions of both
the short and long run investors of immediate or delayed-trading expectations.
In a nutshell, under the expectation of immediate liquidity-trading, long-run investors
expect to obtain the assets of short run investors at close to fair value. In this case the returns
of holding outside liquidity are low and thus there is little of it held by long run investors.
On the other side of the liquidity trade, short run investors will then expect to be able to
sell a relatively small fraction of assets at close to fair value, and therefore respond by relying
more heavily on inside liquidity. That is, they tend to hold a larger fraction of their assets
in liquid securities or cash. In other words, in an immediate trading equilibrium there is less
cash-in-the-market pricing (to borrow a term from Allen and Gale, 1998), which reduces the
return to outside liquidity and therefore its’ supply. The reduced supply of outside liquidity,
in turn, causes financial intermediaries to rely more on inside liquidity and, thus, bootstraps
the relatively high equilibrium price for the assets held by short run investors under immediate
liquidity trading.
2
In contrast, under the expectation of delayed liquidity trading, short run investors rely
more on outside liquidity. Here the bootstrap works in the other direction, as long run investors
decide to hold more cash in anticipation of a larger future supply of the assets held by short
run investors at more favorable cash-in-the-market pricing. The reason why there is more
favorable cash-in-the-market pricing in the delayed trading equilibrium, in spite of the worse
lemons problem, is that in this equilibrium the return to investing in the long maturity asset
is also higher, due to the lower overall probability of liquidating assets at fire-sale prices.
In sum, immediate trading equilibria are based on a greater reliance on inside liquidity
than delayed trading equilibria. And, to the extent that there is a greater reliance on outside
liquidity by short run investors in a delayed trading equilibrium, one should expect – and
we indeed establish – that equilibrium prices of financial intermediary assets are lower in the
delayed-trading than in the immediate-trading equilibrium. In other words, our model predicts
a common dynamic of liquidity crises, in which asset prices progressively deteriorate throughout
the crisis. Importantly, this predictable pattern in asset prices is consistent with no arbitrage,
as short run investors prefer to delay asset sales, despite the deterioration in asset prices, in
the hope that they wont have to trade at all at fire sale prices.1
Because of this deterioration in asset prices one would expect that welfare is also worse in
the delayed-trading equilibrium. However, this is not the case in our model. As it turns out, the
Pareto superior equilibrium is in fact the delayed-trading equilibrium. What is the economic
logic behind this somewhat surprising result? The answer is that the fundamental gains from
trade in our model are between short horizon investors, who undervalue long term assets, and
long horizon investors, who undervalue cash. Thus, the more short horizon investors can be
induced to hold long assets and the more long horizon investors can be induced to hold cash,
the higher are the gains from trade and therefore the higher is welfare. In other words, the
welfare efficient form of liquidity in our model is outside liquidity. Since the delayed trading
equilibrium relies more on outside liquidity it is more efficient.
Under complete-information, when fundamental asset values are fully known to both
short run investors and other investors, it is actually efficient to rely exclusively on outside
liquidity in our model. In the presence of adverse selection, however, outside liquidity involves
too much dilution of ownership and is generally too costly for short run investors, so that they
will tend to rely partially on inside liquidity. As the lemons’ problem worsens – in particular,
1
The short run investors’ decision to delay trading has all the hallmarks of gambling for resurrection. But it
is in fact unrelated to the idea of excess risk taking as these financial intermediaries will choose to delay whether
or not they are levered.
3
as short run investors are less likely to trade for liquidity reasons when they engage in delayed
trading – the cost of outside liquidity rises. There is then a point when the cost is so high
that short run investors and their investors are better off postponing the redemption of their
investments altogether, rather than realize a very low fire-sale price for their valuable longterm assets. At that point the delayed trading equilibrium collapses, as only lemons would get
traded for early redemption.
Interestingly, short run investors could reduce the lemons’ cost associated with outside
liquidity by writing a long-term contract for liquidity with long horizon investors. Such a contract takes the form of an investment fund set up by long horizon investors in which short run
investors can co-invest and in return receive state-contingent payments. Under complete information such a fund arrangement would always dominate any equilibrium allocation achieved
through spot trading of assets for cash. However, when the long-horizon investor who manages
the fund also has private information about the realized returns on the fund’s investments
then, as we show, the long-term contract cannot always achieve a more efficient outcome than
the delayed-trading equilibrium. Indeed, the fund manager’s private information then prevents
the fund from making fully efficient state-contingent transfers to the short run investor, thus
raising the cost of providing liquidity.
Given that neither financial markets nor long-term contracts for liquidity can achieve a
fully efficient outcome, the question naturally arises whether some form of public intervention
may provide an efficiency improvement. There are two market inefficiencies that public policy
might mitigate. An ex-post inefficiency, which arises when the delayed-trading equilibrium
fails to exist, and an ex-ante inefficiency in the form of an excess reliance on inside liquidity. It
is worth emphasizing that a common prescription against banking liquidity crises–namely to
require that banks hold cash reserves or excess equity capital–would be counterproductive in
our model. Such a requirement would only force short run investors to rely more on inefficient
inside liquidity and would undermine the supply of outside liquidity. As we illustrate in an
example, such regulations could push the financial sector out of a delayed-trading equilibrium
into an immediate-trading equilibrium. Interestingly, in that case short run investors may
actually hold liquid assets in equilibrium far in excess of what they are required to hold.
Rather than mandate minimum liquid asset holdings by financial intermediaries, a more
effective policy intervention could be to enforce value-at-risk (VAR) type regulations, which require that short run investors sell or reduce their exposure to risky securities when their overall
VAR is too high. Such regulations would have the effect of forcing financial intermediaries to
rely more on outside liquidity, and could therefore help select a delayed-trading equilibrium.
4
An important novel insight of our analysis is, thus, that VAR type regulations which require
divestitures of assets have the unintended positive effect of shifting the overall reliance of the
financial system on a more efficient form of liquidity supply: outside liquidity.
Another potentially beneficial intervention is a policy of public provision of liquidity by
a central bank, which guarantees a minimum price for assets by lending against collateral.
Such a policy could improve efficiency in our model even though short run investors may
actually never rely on the lending facility in equilibrium, simply by inducing intermediaries to
rely less on inefficient inside liquidity in the first place. That is, such a policy would induce
long-term investors to hold more cash in the knowledge that short run investors rely less on
inside liquidity, and thus help increase the availability of outside liquidity. Far from being a
substitute for privately provided liquidity, a commitment to offer public emergency liquidity
could be a complement and give rise to positive spillover effects on the provision of outside
liquidity. Long-term investors would be prepared to hold more cash in the knowledge that
financial intermediaries rely less on inside liquidity, and if the central bank’s lending terms are
sufficiently punitive, they can then hope to acquire valuable long-term assets at a reasonable
price.
Related literature. Our paper is related to the literatures on banking and liquidity crises,
and the limits of arbitrage. Our analysis differs from the main contributions in these literatures mainly in two respects: first, our focus on ex-ante efficiency and equilibrium portfolio
composition, and second, the endogenous timing of liquidity trading. Still, our analysis shares
several important themes and ideas with these literatures. We briefly discuss the most related
contributions in each of these literatures in turn.
Consider first the banking literature. Diamond and Dybvig (1983)2 provide the first
model of investor liquidity demand, maturity transformation, and inside liquidity. In their
model a bank run may occur if there is insufficient inside liquidity to meet depositor withdrawals. In contrast to our model, investors are identical ex-ante, and are risk-averse with
respect to future liquidity shocks. The role of financial intermediaries is to provide insurance
against idiosyncratic investors’ liquidity shocks.
Bhattacharya and Gale (1986) provide the first model of both inside and outside liquidity
by extending the Diamond and Dybvig framework to allow for multiple banks, which may
face different liquidity shocks. In their framework, an individual bank may meet depositor
withdrawals with either inside liquidity or outside liquidity by selling claims to long-term assets
2
See also Bryant (1980).
5
to other banks who may have excess cash reserves. An important insight of their analysis is
that individual banks may free-ride on other banks’ liquidity supply and choose to hold too
little liquidity in equilibrium.
More recently, Allen and Gale (2000) and Freixas, Parigi, and Rochet (2000) (see also
Aghion, Bolton and Dewatripont, 2000) have analyzed models of liquidity provided through
the interbank market, which can give rise to contagious liquidity crises. The main mechanism
they highlight is the default on an interbank loan which depresses secondary-market prices
and pushes other banks into a liquidity crisis. Subsequently, Acharya (2001) and Acharya and
Yorulmazer (2005) have, in turn, introduced optimal bailout policies in a model with multiple
banks and cash-in-the-market pricing of loans in the interbank market.
While Diamond and Dybvig considered idiosyncratic liquidity shocks and the risk of
panic runs that may arise as a result of banks’ attempts to insure depositors against these
shocks, Allen and Gale (1998) consider aggregate business-cycle shocks and point to the need
for equilibrium banking crises to achieve optimal risk-sharing between depositors. In their
model aggregate shocks may trigger the need for asset sales, but their analysis does not allow
for the provision of both inside and outside liquidity.
Another strand of the banking literature, following Holmstrom and Tirole (1998) considers liquidity demand on the corporate borrowers’ side rather than on depositors’ side, and asks
how efficiently this liquidity demand can be met through bank lines of credit. This literature
emphasizes the need for public liquidity to supplement private liquidity in case of aggregate
demand shocks.
Most closely related to our model is the framework considered in Fecht (2004), which
itself builds on the related models of Diamond (1997) and Allen and Gale (2000). The models of
Diamond (1997) and Fecht (2004) seek to address an important weakness of the Diamond and
Dybvig theory, which cannot account for the observed coexistence of financial intermediaries
and securities markets. Liquidity trading in secondary markets undermines liquidity provision
by banks and obviates the need for any financial intermediation in the Diamond and Dybvig
setting, as Jacklin (1987) has shown. To address this objection, Diamond (1997) introduces
a model where banks coexist with securities markets due to the fact that households face
costs in switching out of the banking sector and into securities markets. Fecht (2004) extends
Diamond (1997) by introducing segmentation on the asset side between financial intermediaries’
investments in firms and claims issued directly to investors though securities markets. Also, in
his model banks have local (informational) monopoly power on the asset side, and subsequently
can trade their assets in securities markets for cash–a form of outside liquidity. Finally, Fecht
6
(2004) also allows for a contagion mechanism similar to Allen and Gale (2000) and Diamond and
Rajan (2005)3 , whereby a liquidity shock at one bank propagates itself through the financial
system by depressing asset prices in securities markets.
Two other closely related models are Gorton and Huang (2004) and Parlour and Plantin
(2007). Gorton and Huang also consider liquidity supplied in a general equilibrium model and
also argue that publicly provided liquidity can be welfare enhancing if the private supply of
liquidity involves a high opportunity cost. However, in contrast to our analysis they do not look
at the optimal composition of inside and outside liquidity, nor do they consider the dynamics
of liquidity trading. Parlour and Plantin (2007) consider a model where banks may securitize
loans, and thus obtain access to outside liquidity. As in our setting, the efficiency of outside
liquidity is affected by adverse selection. But in the equilibrium they characterize liquidity
may be excessive for some banks–as it undermines their loan origination standards–and too
low for other banks, who may be perceived as holding excessively risky assets.
The second literature our model is related to is the literature on liquidity and the dynamics of arbitrage by capital or margin-constrained speculators in the line of Dow and Gorton
(1993) and Shleifer and Vishny (1997). The typical model in this literature (e.g. Kyle and
Xiong, 2001 and Xiong, 2001) also allows for outside liquidity and generates episodes of fire-sale
pricing–even destabilizing price dynamics–following negative shocks that tighten speculators’
margin constraints. However, most models in this literature do not address the issue of deteriorating adverse selection and the timing of liquidity trading, nor do they explore the question of
the optimal mix between inside and outside liquidity. The most closely related articles, besides
Kyle and Xiong (2001) and Xiong (2001) are Gromb and Vayanos (2002), Brunnermeier and
Pedersen (2007) and Kondor (2007). In particular, Brunnermeier and Pedersen (2007) also
focus on the spillover effects of inside and outside liquidity, or what they refer to as funding
and market liquidity.
II. THE MODEL
We consider a model with three phases: an investment phase (date 0), an interim trading
phase (dates 1 and 2) and an unwinding phase of all long duration assets (date 3). There
are two classes of agents which differ in their investment horizons as well as their investment
opportunity sets. In particular, one class of agents is potentially subject to a maturity mismatch
during the interim trading phase and this generates a demand for liquidity. This demand can
3
Another feature in Diamond and Rajan (2005) in common with our setup is the idea that financial inter-
mediaries possess superior information about their assets, which is another source of illiquidity.
7
be met with either cash carried by the agents subject to this maturity mismatch or by the
sale of assets to the other class of agents, who may also carry cash to acquire these assets
opportunistically. We call the cash carried by those who demand liquidity, inside liquidity,
and the cash supplied by the second class of agents to acquire the assets outside liquidity. The
novelty of our analysis resides in the timing of these asset sales. As mentioned, our interim
trading phase is divided in two distinct periods. In the first, date 1, there is an aggregate
shock that determines the average quality of the assets held by one class of agents and that
is known to all. In date 2 there are idiosyncratic shocks to the assets and knowledge of the
nature of these shocks accrues only to those in possession of the assets. Thus trading in this
last date occurs under conditions of asymmetric information. Our purpose is to understand
what determines the timing of the asset sales and the distribution of outside versus inside
liquidity. We also want to understand the welfare consequences associated with these choices
and thus offer regulatory prescriptions in case the market does not produce efficient outcomes.
After this brief sketch of the model we now give a more detailed account of the framework.
II.A Agents
There are two types of agents, short and long run investors with preferences over periods
t = 1, 2, 3. Short run investors, of which there is a unit mass, have preferences
u(C1 , C2 , C3 ) = C1 + C2 + δC3 ,
(1)
where Ct ≥ 0 denotes consumption at dates t = 1, 2, 3 and δ ∈ (0, 1). These investors have
one unit of endowment at date0 and 0 in every other date. There are also long run investors:
There is a unit mass of long run investors, each with κ > 0 units of endowment at t = 0 and
again no endowment at subsequent dates. Their utility function is simply given by
X3
û(C1 , C2 , C3 ) =
Ct ,
t=1
provided Ct ≥ 0.
In what follows we refer to short and long run investors as SRs and LRs respectively.
II.B Assets
For simplicity we assume that the two types of investors have access to different investment opportunity sets and comment on this assumption further below. First, both types can
hold cash with a gross per-period rate of return of one. LRs in addition can invest in a long
asset. Specifically, we assume that each LR has access to a decreasing-returns-to-scale long
maturity asset that returns ϕ(x) at date 3 for an initial investment at date 0 of x = (κ − M ),
where M ≥ 0 is the LRs’ cash holding and our definition of outside liquidity.
8
SRs can invest in a risky asset, which is a constant returns to scale technology, that pays
an amount e
ρt at dates t = 1, 2, 3 where e
ρt ∈ {0, ρ} and ρ > 1. The payoff of the risky asset is
the only source of uncertainty in the model and is shown in Figure 1. For simplicity we assume
that there is a first aggregate maturity shock that affects all risky assets. That is, agents learn
first whether all risky assets mature at date 1, or at some later date. Subsequently, the realized
value of a risky asset and whether it matures at date 2 or 3 is determined by an idiosyncratic
shock.
Formally, the risky asset either pays ρ at date 1 (in state ω 1ρ ), which occurs with probability λ, or it pays at a return at a subsequent date, with probability (1 − λ). In that case
the asset yields either a return e
ρ2 ∈ {0, ρ} at date 2, or a late return e
ρ3 ∈ {0, ρ} at date 3.
After date 1 shocks are idiosyncratic and are represented by two separate i.i.d. random variables: (i) an individual asset can either mature at date 2, with probability θ, or at date 3 with
probability (1 − θ) (in state ω 2L ); (ii) when the asset matures at either dates t = 2, 3 it yields
e
ρt = ρ with probability η (in states ω 2ρ and ω 3ρ , respectively) and e
ρt = 0 with probability
(1 − η) for t = 2, 3 (in states ω 20 and ω 30 .) The realization of the idiosyncratic shocks is private
information to the SR holding the risky asset. We denote by m the amount of cash held by
SRs and then 1 − m is the amount invested in the risky asset. m is thus our measure of inside
liquidity.
II.C Financial markets
At dates 1 and 2 a secondary market opens where claims on the SR’s risky asset can
be traded for the cash held by LRs. In particular SRs can liquidate the risky assets in state
ω 1L , which we refer to as the immediate trading date. Alternatively SRs can instead postpone
decisions until date 2 and thus “give another chance” for the asset to pay off. Recall that
knowledge of the idiosyncratic shocks accrues only to SRs and thus only they know whether
they are in states ω 2ρ , ω 20 , or ω 2L . SRs in state ω 2ρ collect the payoff and consume accordingly.
SRs in state ω 20 have worthless assets whereas SRs in ω 2L have assets with an expected payoff
of ηρ, a payoff that is realized at date 3. SRs in ω 2L can either liquidate the asset or carry it
to date 3. LRs instead only know that the assets sold at date 2 can either be “lemons,” those
sold by SRs in state ω 20 , or good assets which can still pay at date 3.
II.D Assumptions
We introduce some minimal assumptions that will focus the analysis on the economically
interesting questions and simplify considerably the presentation. We start with assumptions
9
on the different technologies. First we assume that outside liquidity is costly:
ϕ0 (κ) > 1
with
ϕ00 (x) < 0 and
lim ϕ0 (x) = +∞
x−→0
(A1)
The assumption that ϕ00 (·) < 0 captures the the fact that the opportunities that these long
assets represent are scarce and cannot be exploited limitlessly. To simplify the presentation we
also assume that LRs always want to invest in this long asset, that is, that limx→0 ϕ0 (x) = +∞.
The key assumption here though is that ϕ0 (κ) > 1. Thus if LRs carry cash it must be to
acquire, in some states of nature, assets with high expected returns. Given the assumption of
risk neutrality this can only occur if asset purchases occur at cash-in-the-market prices. That
is, as we will show more formally below, assets trade at prices that are below the expected
payoff, for otherwise LRs would have no incentive to carry cash.
Our second assumption says that SRs would not invest in the risky asset in autarchy,
though investment in it is more attractive than cash when the asset can be resold:
ρ [λ + (1 − λ)η] > 1
and
λρ + (1 − λ) [θ + (1 − θ) δ] ηρ < 1
(A2)
Assumption A2 is needed to get the economically interesting situation where the liquidity
of secondary markets at dates 1 and 2 affects asset allocation decisions at date 0. If instead
we assumed that
λρ + (1 − λ) [θ + (1 − θ) δ] ηρ ≥ 1
then SRs would always choose to put all their funds in a risky asset irrespective of the liquidity
of the secondary market at date 1.
Finally we assume that there are gains from trade, at least, in the immediate trading
date, at date 1. That is, ϕ0 (κ) is not as high as to rule out the possibility of the LRs carrying
cash to trade at date 1 altogether. As will become clear below, assumption A3 states that the
agents’ isoprofit lines cross in the right way:
ϕ0 (κ) − λ
1−λ
<
(1 − λ) ηρ
1 − λρ
(A3)
II.E Discussion
A central feature of the model is the timing of the aggregate and idiosyncratic shocks.
The aggregate shock reveals news about the entire class of risky assets, whether they pay off,
as it occurs in state ω 1ρ , or instead whether there is a deterioration of the entire class as well
as postponement of the actual payoff, as it occurs in state ω 1L . Indeed, the expected payoff of
10
the risky assets in state ω 1L is ηρ, and the payoff may or not be realized at date 2 or3. After
date 1, additional news accrue only to the holder of the asset and that is what generates the
adverse selection premium. In state ω 1L all agents are informed about problems in a particular
financial market, which results in a drop in prices. The key is that some of the holders of the
risky asset are indeed affected and others are not, but they themselves only learn of which
state they are in at t = 2. It is for this reason that if transactions take place at date 1 in state
ω 1L prices do not include an adverse selection premium whereas they do at date 2.
Once the initial portfolios are set, a critical trade-off the agents face in our framework
is the decision of whether to liquidate and acquire the assets at date 1 or 2. The parameter θ
plays an important role in determining this decision. Indeed a high value of θ means that if
the risky asset has not paid off at date 1, the probability that it does so at date 2 rather than
at date 3 is also high. This makes the risky asset attractive to the SRs for they care about
consumption at date 2 more than they do about consumption at date 3. But at the same time,
the higher the value of θ, the more severe the adverse selection problems are at date 2 because
the higher the θ the higher the probability that the asset acquired is a lemon. This translates
into an adverse selection premium that discourages the SRs from carrying the asset to date 2.
Similarly for the LRs the trade-off is between acquiring high quality assets at date 1 at high
prices or trade in a market subject to adverse selection but potentially at better prices. How
these trade-offs affect the (ex-ante) portfolio decisions of both SRs and LRs is the central issue
explored in this paper.
We finish this section with some comments about our particular assumptions on the
contractual framework the agents face. First we assume that SRs cannot invest in the long
asset and LRs cannot invest in the risky asset. Clearly, because limx→0 ϕ0 (x) = +∞, SRs
would want to invest in the long asset at least a small amount, but recall that their marginal
utility of consumption is given by δ, which we assume to be “small,” in a sense we make precise
below, and thus SRs would still be faced with a trade-off between the risky asset and cash. Also
we assume that the LRs cannot invest in the risky asset, but this turns out to be less relevant
for our purposes than it may appear at first. Indeed the risky asset is a constant returns to
scale technology. If the returns to holding cash, what we refer to as outside liquidity, are below
the expected return of the risky asset for the LRs, which is ρ [λ + (1 − λ) η], then the LRs
would not want to carry any cash and by assumption A2, the only resulting equilibrium would
be one where the SRs would not invest in the risky asset. If instead the returns to holding
cash are above the risky asset’s expected return then clearly the LRs would not invest in it.
Clearly the first situation is not an interesting one in which to analyze the determinants of
11
outside versus inside liquidity and it is for this reason that we simply assume that the LRs
cannot invest in the risky asset.
Finally a word about the endogeneity of the outside capital that can potentially absorb
the risky asset sales by part of SRs. In principle the entire supply of capital in the economy
should stand ready to absorb these sales the moment returns are attractive enough, which
realistically should yield small quantitative effects for the types of problems discussed in this
paper. But we argue that, as the literature on limits to arbitrage postulates, sometimes the
capital and the knowledge of the particular good opportunities in a market are unbundled,
either because the capital of those with knowledge has been depleted due to some adverse
shocks or simply because moral hazard and adverse selection problems are preventing capital
from flowing to the market with the apparent attractive opportunities. Understanding the
frictions that lead to this unbundling is essential to have a complete picture of financial markets
and it is the subject of our current research. In this paper we simply take this amount of outside
capital, κ, that can potentially absorb the SR’s sales to be a parameter, rather than a variable.
III. EQUILIBRIUM
Given that all SRs and LRs are ex-ante identical, we shall restrict attention to symmetric
competitive equilibria. Recall that trade between SRs and LRs can only occur in spot markets
at date 1 and/or 2, and there are only two information sets at which there are potentially strictly
positive gains from trade, ω 1L and {ω 20 , ω 2L }. Given that SRs have private information about
realized returns on their risky asset at date 2, they can condition their trading policy on states
ω 1L , ω 20 and ω 2L . LR investors, on the other hand, are unable to distinguish states ω 20 and
ω 2L , and therefore can only condition their trades on their information sets ω 1L and {ω 20 , ω 2L }.
We denote by q (ω 1L ) be the amount of the risky asset supplied by an SR in state ω 1L and by
q (ω 20 , ω 2L ) the amount supplied at date 2. Similarly, we denote by Q (ω 1L ) (Q (ω 20 , ω 2L )) the
amount of the risky asset that LR investors acquire in ω 1L ({ω 20 , ω 2L }).
III.A The SR optimization problem
SRs must determine first how much of an investor’s savings to invest in cash and how
much in a risky asset. Second, they must decide how much of the risky asset to trade at date
1 at price P1 and at date 2 at price P2 . Their objective function is then
π [m, q (ω 1L ) , q (ω 20 , ω 2L )] = m + λ (1 − m) ρ
+ (1 − λ) q (ω 1L ) P1
12
+ (1 − λ) θη [(1 − m) − q (ω 1L )] ρ
+ (1 − λ) θ (1 − η) [(1 − m) − q (ω 1L )] P2
(2)
+ (1 − λ) (1 − θ) q (ω 20 , ω 2L ) P2
+ δ (1 − λ) (1 − θ) η [(1 − m) − q (ω 1L ) − q (ω 20 , ω 2L )] ρ
Notice that implicit in this objective function is the fact that SRs don’t trade in states
ω 1ρ , ω 2ρ and ω 3ρ . As we have noted above, given SRs’ preferences and the LRs’ objective
function below there is actually no gain from trading assets in these states of nature.4 In state
ω 1ρ , which occurs with probability λ, the risky asset pays in full at date 1 and SRs consume
all the proceeds. In contrast, in state ω 1L the risky asset matures at a later date, and SRs
may choose to sell an amount q (ω 1L ) of the risky asset for a unit price P1 .
The risky asset then matures with ex-ante probability (1 − λ) θη at date 2, in which
case SRs consume the share of the proceeds of the asset it still owns: [(1 − m) − q (ω 1L )]ρ.
Another outcome at date 2 is that the asset yields a zero return. This occurs with probability
(1 − λ)θ (1 − η). In that state of nature the SR chooses optimally to sell its full position in the
risky asset (which is now a lemon) for the price P2 . Finally, with probability (1 − λ)(1 − θ) the
asset only matures at date 3. The SR may then sell an additional amount of the risky asset
q (ω 20 , ω 2L ) at unit price P2 rather than holding it to maturity at date 3, when the asset yields
a return ρ with ex-ante probability (1 − λ)(1 − θ)η. The SR’s optimal investment program is
therefore given by:
max
m,q(ω 1L ),q(ω 2L )
π [m, q (ω 1L ) , q (ω 20 , ω 2L )]
(PSR )
subject to
m ∈ [0, 1]
and
q (ω 1L ) + q (ω 20 , ω 2L ) ≤ 1 − m with q (ω 1L ) , q (ω 20 , ω 2L ) ≥ 0
The constraints simply state that the SR cannot invest more in the risky asset than the
funds at its disposal and that in states ω 1L and (ω 20 , ω 2L ) it cannot sell more than what it
holds.
III.B The LR optimization problem
LRs must first determine how much of their savings to hold in cash, M , and how much
in long term assets, κ − M . They must then decide at dates 1 and 2 how much of the risky
4
Recall that the marginal utility of consumption at date t = 3 is δ ∈ (0, 1).
13
assets to purchase at prices P1 and P2 . Recall that, given assumption 2 cash is costly to carry
for LRs and thus they never carry more cash than they expect they will need to purchase risky
assets from SRs at dates 1 and 2. In other words, in the states of nature where trade occurs LR
investors completely exhaust their cash reserves to purchase the available supply of SR long
assets. With this observation in mind we can write the payoff an LR investor that purchases
Q (ω 1L ) at date 1 and Q (ω 20 , ω 2L ) at date 2, as follows:
Π [M, Q (ω 1L ) , Q (ω 20 , ω 2L )] = M + ϕ (κ − M )
+ (1 − λ) [ηρ − P1 ] Q (ω 1L )
(3)
+ (1 − λ)E [ρ̃3 − P2 | F]Q (ω 20 , ω 2L )
The first term in the above expression is simply what the LR investor gets by holding an
amount of cash M until date 3 without ever trading in secondary markets at dates 1 and 2.
The second term is the net return from acquiring a position Q(ω 1L ) in risky assets at unit
price P1 at date 1. Indeed, the expected gross return of a risky asset in state ω 1L is ηρ. The
last term is the net return from trading in states (ω 20 , ω 2L ). This net return depends on the
payoff of the risky asset at date 3 and in particular on the quality of assets purchased at date
2. As we postulate rational expectations, the LR investor’s information set, F, will include the
particular equilibrium that is being played. In computing conditional expectations the LRs
assume that the mix of assets offered in states (ω 20 , ω 2L ) corresponds to the one observed in
equilibrium.
We require a standard, and weak, rationality condition from LRs, that if they succeed in
purchasing a unit in states (ω 20 , ω 2L ) in an equilibrium that prescribes no sales in these states,
and furthermore at a price for which SRs strictly prefer to hold the asset until date 3 to selling
it in state ω 2L , then LR assumes that state ω 20 (where SRs always prefer to sell) has occurred
for sure. In addition, LRs assume that SRs that weakly prefer to sell at price P2 will sell all
their remaining holdings in states (ω 20 , ω 2L ) whereas SRs that prefer not to sell, will not sell
any units.
The LR investor’s program is thus:
max
M,Q(ω 1L ),Q(ω 20 ,ω 2L )
Π [M, Q (ω 1L ) , Q (ω 20 , ω 2L )]
(PLR )
subject to
0≤M ≤κ
(4)
and
Q (ω 1L ) P1 + Q (ω 20 , ω 2L ) P2 ≤ M
14
with Q (ω 1L ) , Q (ω 20 , ω 2L ) ≥ 0
(5)
The first constraint (4) is simply the LR investor’s wealth constraint: LRs’ cannot carry
more cash than their initial capital κ and they cannot borrow. The second constraint (5) says
that LRs’ cannot purchase more SR long-assets than their money, M , can buy. In our model
M is, thus, the supply of outside liquidity by LRs.
III.C Definition of equilibrium
A (non-revealing) rational expectations competitive equilibrium is a vector of portfolio
policies [m∗ , M ∗ ], supply and demand choices [q ∗ (ω 1L ) , q ∗ (ω 20 , ω 2L ) , Q∗ (ω 1L ) , Q∗ (ω 20 , ω 2L )]
and prices [P1∗ , P2∗ ] such that (i) at these prices [m∗ , q ∗ (ω 1L ) , q ∗ (ω 20 , ω 2L )] solves PSR and
[M ∗ , Q∗ (ω 1L ) , Q∗ (ω 20 , ω 2L )] solves PLR and (ii) markets clear in all states of nature.
III.D Characterization of equilibria
III.D.1 Immediate and delayed-trading equilibria
The immediate-trading equilibrium. Under our stated assumptions we are able to establish first that there always exists an immediate trading equilibrium.
Proposition 1. (The immediate-trading equilibrium) Assume A1-A3 hold then there always
exists an immediate trading equilibrium, where
Mi∗ > 0, q ∗ (ω 1L ) = Q∗ (ω 1L ) = 1 − m∗i
q ∗ (ω 20 , ω 2L ) = Q∗ (ω 20 , ω 2L ) = 0.
and
In this equilibrium cash-in-the-market pricing obtains and
∗
P1i
=
Mi∗
1 − λρ
≥
.
1 − m∗i
1−λ
(6)
Moreover the cash positions m∗i and Mi∗ are unique.
To gain some intuition start by noticing that the immediate trading equilibrium has to
meet the corresponding first order conditions for m and M , which are
∗
≥
P1i
1 − λρ
1−λ
and
λ + (1 − λ)
ηρ
0
∗
∗ = ϕ (κ − Mi ) ,
P2i
(7)
when m∗i < 1 and Mi∗ > 0,5 as it follows from immediate inspection of the maximization
problems PSR , once we assume q ∗ (ω 1L ) = 1 − m∗i , and PLR , respectively. The question is how
5
The proof of Proposition 1 establishes that assumption A3 rules out the possibility of a “no trade” immediate
trading equilibrium in which Mi∗ = 0 and m∗i = 1.
15
∗ . For this consider P ∗ to be the unique solution to:
to determine P1i
1i
λ + (1 − λ)
ηρ
= ϕ0 (κ − P1i ) .
P1i
(8)
Assume first that the solution to (8) is such that
P1i >
1 − λρ
,
1−λ
then we can set m∗i = 0, and thus the SR is fully invested in the risky asset, and also set
∗ = M ∗ , which by construction satisfies the LR’s first order condition and because of A1 has
P1i
i
to be such that Mi∗ < κ. A key step in the construction of the immediate trading equilibrium
∗ , has to be such that both SRs and LRs have incentives to trade
is that the price at date 2, P2i
at date 1 and not at date 2. That is, it has to be that
∗
∗
P1i
≥ θηρ + (1 − θη) P2i
and
ηρ
E [e
ρ3 |F]
≥
.
∗
∗
P1i
P2i
(9)
∗ rather
The first expression in (9) states that SRs prefer to sell assets at date 1 for a price P1i
than carrying it to date 2. If they do the latter, with probability θη the risky asset pays off
ρ. With probability (1 − θη) instead they end up in states (ω 2L , ω 20 ) and the SRs can sell
∗ . If the price P ∗ is low enough,6 then indeed the SRs would prefer to
the asset at price P2i
2i
liquidate at date 1.
The expression on the right hand side of (9) states that the expected return of acquiring
the asset in state ω 1L is higher than in states (ω 20 , ω 2L ). To guarantee this outcome it is
∗ < δηρ for in this case SRs in state ω
sufficient to set P2i
2L would prefer to carry the asset to
date 3 rather than selling it for that price. This leaves only “lemons” in the market. LRs,
anticipating this outcome, set their expectations accordingly, E [e
ρ3 |F] = 0, and thus for any
∗ < δηρ LRs prefer to acquire assets in state ω .
strictly positive price P2i
1L
Assume now that instead the solution to (8) is such that
P1i ≤
1 − λρ
,
1−λ
(10)
∗ equal to the right hand side of (10). At this price, SRs are indifferent on the
then set P1i
amount of cash carried. Then the solution to the LR’s first order condition is such that (see
expression (7)):
Mi∗ <
6
1 − λρ
.
1−λ
∗
See expression (25) in the appendix for a precise upper bound on P2i
that has to hold to provide incentives
for the SRs to sell at date 1 rather than at date 2.
16
In which case it is sufficient to set m∗i ∈ [0, 1) such that:
Mi∗
1 − λρ 7
=
,
1 − m∗i
1−λ
(11)
∗ is as above.
which is always possible. Finally, the choice of P2i
Notice that in our framework, and by assumption A1, cash in the market has to obtain
∗ < ηρ, otherwise there would
and prices are lower than their discounted expected payoff, P1i
be no incentive for the LRs to carry cash.
The delayed-trading equilibrium. Proposition 2 establishes the existence of a delayed
trading equilibrium.
Proposition 2.8 (The delayed-trading equilibrium) Assume A1-A3 hold and that δ is small
enough9 then there always exists an delayed-trading equilibrium, where m∗d ∈ [0, 1),
Md∗ ∈ (0, κ),
q ∗ (ω 1L ) = Q∗ (ω 1L ) = 0
and
q ∗ (ω 20 , ω 2L ) = Q∗ (ω 20 , ω 2L ) = (1 − θη) (1 − m∗d ) .
In this equilibrium cash-in-the-market pricing obtains and
∗
P2d
=
Md∗
1 − ρ [λ + (1 − λ) θη]
¢≥
¡
.
∗
(1 − λ) (1 − θη)
(1 − θη) 1 − md
(12)
Moreover the cash positions m∗d and Md∗ are unique.
The intuition of the construction of the delayed-trading equilibrium is broadly similar
to the immediate-trading equilibrium, with some differences that we emphasize next. First,
as stated in the proposition δ needs to be small enough. Otherwise SRs in state ω 2L prefer
to carry the asset to date 3 rather than selling it at date 2, which would destroy the delayedtrading equilibrium as it would only leave lemons in the market. We postpone a discussion of
the existence problems that arise when δ is relatively high until later in the paper.
Second, a key difference with immediate trading is that the supply of risky assets by SRs
is reduced under delayed trading by an amount θη, which is the proportion of risky assets that
7
8
Notice that assumption A2 implies that 1 − λρ > 0
There may also be a third equilibrium, which involves positive asset trading at both dates 1 and 2. We do
not focus on this equilibrium as it is unstable.
9
The proof of the proposition clarifies the upper bound on δ that guarantees existence, see expression (36)
in the Appendix.
17
pay off at date 2.10 Cash-in-the-market pricing is now:
∗
P2d
=
Md∗
¡
¢.
(1 − θη) 1 − m∗d
Notice that now the mass of risky assets supplied in the market in states (ω 20 , ω 2L ) is given by
(1 − θη) (1 − m∗d ). Thus delaying asset liquidation introduces an adverse selection effect, which
depresses prices, and a lower supply of the risky asset, which, other things equal, increases
prices.
As before supporting a delayed-trading equilibrium requires that both SRs and LRs have
incentives to trade at date 2 rather than at date 1, which means that
∗
∗
P1d
≤ θηρ + (1 − θη) P2d
and
ηρ
E [e
ρ3 |F]
≤
,
∗
∗
P1d
P2d
(13)
where now the expected payoff is given by
E [e
ρ3 |F] =
(1 − θ)ηρ
.
1 − θη
If (13) is to be met, the price in state ω 1L has to be in the interval,
·
¸
1 − θη ∗
∗
∗
P1d
∈
P2d , θηρ + (1 − θη)P2d
.
1−θ
The key step of the proof of Proposition 2 is that this interval is non empty.
It is perhaps worth emphasizing that the delayed trading equilibrium collapses to the
immediate trading equilibrium when θ = 0. Indeed notice, for instance, that the lower bound
∗ in (12) reduces to the lower bound in (6) for P ∗ . The only difference between
in the price P2d
1i
dates 1 and 2 is precisely the occurrence of an idiosyncratic shock that reveals to the SRs, the
holders of the risky asset, its true value. When θ = 0 the idiosyncratic signal does not occur
and thus all the information is common knowledge as it is always the case at date 1. This
feature of our model will play an important role in what follows.
Before we close this section we introduce the example that we use throughout to illustrate
the results. Because as already explained, the parameter θ plays a critical role in our analysis
it is the focus of the comparative statics below and thus we parameterize the set of economies
that we considered throughout by the different values that it takes. In light of assumption A2
it is then convenient to define θ as the value for which
£
¡
¢¤
1 = ρ λ + (1 − λ)ηρ θ + (1 − θ)δ ,
10
(14)
This is one of the key differences that arises when the shocks at date t = 2 are aggregate rather than
idiosyncratic. In this case the supply of risky assets is always the same. The difference is that there is one
aggregate state of nature, ω 1ρ where there is no mare for the risky asset at date t = 2.
18
for a given λ, δ, η, and ρ.
Example. In this example the parameter values are:
λ = .85
η = .4
ρ = 1.13
κ = .2
δ = .1920
ϕ (x) = xγ
with γ = .4
Having fixed the value of δ, we need to restrict the values of our only free parameter θ to
θ ≤ θ = .4834 so as to ensure that assumption A2 holds. It is immediate to check that
in this example assumptions A1-A3 hod, as well as assumption A4 below. Figures and
numbers throughout the paper refer to this example. A summary of the main features
follows.
• Both the immediate and delayed trading equilibria exist for θ ∈ [0, .4196) and
moreover in the delayed trading equilibrium we have m∗d > 0.
• For θ ∈ [.4196, .4628] both equilibria exist and the delayed trading equilibrium is
such that m∗d = 0.
• For θ ∈ (.4628, .4834] the delayed trading equilibrium fails to exists. It is for this
range that the assumption of δ being “small enough” fails to hold and we elaborate
on this case below.
2
III.D.2 Inside and outside liquidity in the immediate and delayed trading equilibria
How does the composition of inside versus outside liquidity vary across equilibria? To
build some initial intuition on this question it is useful to illustrate the immediate and delayedtrading equilibrium that obtain in our example when θ = .35. Figure 2 represents the immediate and delayed-trading in a diagram where the x axis measures the amount of cash carried
by LRs, M , and the y axis the amount of cash carried by the SRs, m. The dashed lines are
the isoprofit curves of the LRs and the straight (continuous) lines are the SR isoprofit lines.11
In the figure we have both the isoprofit lines for both the immediate and delayed exchange,
which of course, correspond to exchange of different assets as the one at date 2 has different
payoffs than the one at date 1.12 It is for this reason that isoprofit lines appear to cross in the
plot: They simply correspond to different dates. Equilibria are located at the tangency points
between the isoprofit lines of the SRs and LRs.
11
Assumption A3 simply says that the slope of the isoprofit lines at M = 0 in the immediate-trading date are
such that there ganis from trade: The LR’s isoprofit line is “flatter” than the SR’s isoprofit line.
12
Figure 2 is simply the result of superimposing two Edgeworth Boxes, the one corresponding to the immediate
exchange and the one corresponding to the delayed exchange.
19
Start with the immediate trading equilibrium, located at the point marked (Mi∗ , m∗i ) =
(.0169, .9358)). There are two isoprofit lines going through that point; the straight line corresponds to the SR and the dashed-dotted line corresponds to the LR’s isoprofit line. In fact
the straight line corresponds to the SR’s reservation utility, π = 1. Thus whatever gains from
trade there are in the immediate trading equilibrium they accrue to the LRs. Turn next to
the delayed-trading equilibrium, which is marked (Md∗ , m∗d ) = (.0540, 4860) and features a mix
of outside versus inside liquidity that is tilted towards the former relative to the latter when
compared to the immediate-trading equilibrium. Also, observe that the SR’s isoprofit line has
shifted down, reflecting the fact that the perceived “quality” of SR assets in states (ω 20 , ω 2L ) is
lower than in state ω 1L due to adverse selection, so that one should expect that the SRs would
have to settle for a lower price in that state. The SR’s isoprofit line remains that associated
with it’s reservation value.
One way of understanding the portfolio choices in the immediate-trading equilibrium is
that the risky asset is of high quality in state ω 1L , so that SRs must be compensated with a
high price relative to the price in states (ω 20 , ω 2L ), which also includes an adverse selection
discount, to be willing to sell the asset at that point. This observation is reflected in the slope
of the isoprofit lines in Figure 2: The SRs’ isoprofit line in the immediate trading equilibrium is
flatter suggesting that SRs require a higher price per unit of risky asset sold at that date. But
this higher price can only come at the expense of lower returns to holding cash for LRs. The
latter are thus induced to cut back on their cash holdings. This, in turn, makes it less attractive
for SRs to invest in the risky asset, and so on. The outcome is that in the immediate trading
equilibrium most of the liquidity is inside liquidity held by SRs, whereas the delayed-trading
equilibrium features relatively more outside liquidity than inside liquidity.
The next proposition formalizes this discussion, specifically, it characterizes the mix of
inside versus outside liquidity across the two types of equilibria. For this we make one additional
assumption that allows for a particularly clean characterization of the aforementioned mix,
1 − λρ
>κ
1−λ
(A4)
As the Result in the Appendix shows under assumption A4 the immediate-trading equilibrium is such that m∗i ∈ (0, 1), that is the SRs is carrying a strictly positive amount of cash.
Roughly, we need to guarantee that m∗i > 0 in order to obtain non trivial cash allocation
decisions for the SRs, which otherwise would be equal to 0 for both the immediate and the
delayed-trading equilibria, as will become clear in Proposition 4. The present paper is concerned with the ex-ante efficiency costs associated with portfolio choices that result in the
20
particular timing of the liquidation decisions and thus the most economically interesting case
is the one where the economy is not “at a corner,” that is m∗i = 0, at the immediate-trading
date. Armed with this new assumption we can prove the following
Proposition 3. (Inside and outside liquidity across equilibria.) Assume that A1-A4 hold
and that δ is small enough so that a delayed trading equilibrium exists for all θ ∈ (0, θ]
¡ ¤
then there exists a θ0 ∈ 0, θ such that m∗i > m∗d and Mi∗ < Md∗ for all θ ∈ (0, θ0 ].
Thus for the range θ ∈ [0, θ0 ] the delayed-trading equilibrium features more outside
liquidity and less inside liquidity than the immediate-trading equilibrium. In our example
θ0 = θ so that Proposition 3 holds for the entire range of admissible θs.13
We close this section by making two additional comments. First, note that all equilibria
are interim efficient. That is, conditional on trade occurring in either states ω 1L or (ω 20 , ω 2L ),
there is no additional reallocation of the risky asset that would make both sides better off.
As can be seen immediately in Figure 2, it is not possible to improve the ex-post efficiency of
either equilibrium, as in each case the equilibrium allocation is located at the tangency point of
the isoprofit curves. As we shall further explore below, in our model inefficiencies arise through
distortions in the ex-ante portfolio decisions of SRs and LRs and through the particular timing
of liquidity trades they give rise to. When agents anticipate trade in state ω 1L , SRs lower their
investment in the risky asset and carry more inside liquidity mi . In contrast LRs, carry less
liquidity Mi as they anticipate fewer units of the risky asset to be supplied in state ω 1L .
A second observation is that A4, which implies that the immediate trading equilibrium
is such that m∗i > 0, does not necessarily imply that m∗d > 0. Indeed, Figure 3 shows the
immediate and delayed-trading equilibrium when θ is increased from θ = .35, as it was the
case in Figure 2, to θ = .45 and thus the adverse selection problem is relatively worse than
in the previous case. The delayed-trading equilibrium is located in (Md∗ , m∗d ) = (.0716, 0),
the immediate-trading equilibrium being unaffected as it is independent of θ. Clearly the
equilibrium is ex-post efficient, but now, unlike in the case considered in Figure 2, gains from
trade do not solely accrue to the LRs but also to the SRs. In Figure 3 the isoprofit line marked
IPSR corresponds to the profit level π = 1 for the SR, which is the same as under autarky.
The isoprofit line through the delayed trading equilibrium lies strictly to the right of IPSR ,
which implies that FIs now command strictly positive profits. The reason is that at the corner
when m = 0, FIs are at full capacity in supplying the risky asset in states (ω 20 , ω 2L ). They
13
In fact though we have been unable to prove it formally, we have not found an example of an economy that
meets assumptions A1-A4 for which θ0 < θ.
21
may then earn scarcity rents, as LRs compete for the limited supply of the risky asset supplied
by the SRs by increasing their bids for these assets.
III.D.3 Adverse selection and the delayed trading equilibrium
We now turn to the relevant comparative statics in our analysis, namely how changes in
the adverse selection problem SRs face in states (ω 20 , ω 2L ), as measured by changes in θ, affect
equilibrium outcomes. In particular, we are interested in understanding how equilibrium cash
holdings and equilibrium prices vary with θ.
Several important effects are at work as θ changes, some of which we have already
mentioned. First, the incentives of both SRs and LRs to hold cash are affected by changes in
θ. In addition, SRs’ incentives to hold onto their asset position until date 2 (when the risky
asset does not mature at date 1) are affected. As θ rises the risky asset is more likely to mature
at date 2 and thus becomes more attractive to SRs. Other things equal, SRs are then both
more likely to invest in the risky asset and to carry the asset from date 1 to date 2.
However, as θ rises the adverse selection problem in states (ω 20 , ω 2L ) is worsened and
∗ are likely to be lower. These lower prices that SRs face in
therefore equilibrium prices P2d
states (ω 20 , ω 2L ) in turn reduce their incentives to invest in the risky asset and to carry it to
date 2. An additional effect that complicates the analysis is that as θ increases the supply of
the risky asset in states (ω 20 , ω 2L ),
s∗d (ω 20 , ω 2L ) ≡ (1 − m∗d (θ)) (1 − θη)
(15)
diminishes on account of the fact that a larger share of the available risky assets pay off and
thus are not liquidated.
We are interested also in the expected return on acquiring the risky asset at date 2 in
the delayed-trading equilibrium, which is defined as
Rd∗ (ω 20 , ω 2L ) ≡
(1 − θ) ηρ
∗
(1 − θη) P2d
(16)
The next proposition establishes how these countervailing effects net out and how Md∗ ,
m∗d , s∗d (ω 20 , ω 2L ), and Rd∗ (ω 20 , ω 2L ) vary with θ. Throughout we assume, of course, that θ ≤ θ.
Proposition 4. (Comparative statics.) Assume that A1-A4 hold and that δ is small enough
for all θ ∈ [0, θ] so that a delayed trading equilibrium always exists, then there exists a
unique b
θ ∈ [0, θ], possibly b
θ = θ, such that:
22
I. The SR’s cash position: (a) m∗d is a (weakly) decreasing function of θ, (b) m∗d > 0
for all θ ∈ [0, b
θ) m∗d = 0 for all θ ∈ [b
θ, θ] and (c) s∗d is a strictly increasing function
of θ for θ ∈ [0, b
θ) and a strictly decreasing function of θ for θ ∈ [b
θ, θ].
II. The LR’s cash position: Md∗ is a strictly increasing function of θ for θ ∈ [0, b
θ) and
a strictly decreasing function of θ for θ ∈ (b
θ, θ].
III. Expected returns at date 2: Rd∗ is an increasing function of θ for θ ∈ [0, b
θ) and a
decreasing function of θ for θ ∈ (b
θ, θ].
We illustrate the comparative statics described in Proposition 3 in our example, for
which, given our parametric assumption, it can be shown that b
θ = .4196. Figures 4 and 5
exhibit the comparative statics with respect to θ for the cash positions, m∗d and Md∗ , and the
∗ ),
expected return and the price of the risky asset in states (ω 20 , ω 2L ) (Rd∗ (ω 20 , ω 2L ) and P2d
respectively.
Consider first Figure 4. As we would expect, based on our discussion above, the amount
of cash carried by the SR is a decreasing function of θ, and m∗ = 0 for θ ≥ b
θ = .4196. It is less
d
obvious how the amount of cash carried by LR investors varies with θ. Consider first the case
where θ ≤ b
θ. The amount of cash carried by LR investors is then an increasing function of θ.
This is surprising: the more severe the adverse selection problem the more cash LR investors
bring to states (ω 20 , ω 2L ). What is the logic behind this result?
In this range of θs there are actually two effects at work. First, an increase in θ does
indeed worsen the adverse selection problem and would other things equal result in LRs reducing their supply of liquidity. But there is a countervailing effect, which is that an increase in θ
also results in a higher investment in the risky asset by the SRs. Indeed as shown in 1-(c) in
the proposition, s∗d is an increasing function of θ in this range. It is this higher supply of the
risky asset that in turn increases the supply of outside liquidity. The latter effect dominates
and thus results in an increasing M ∗ as a function of θ when θ ≤ b
θ. Instead, when θ > b
θ the
d
supply effect gets reversed and
s∗d
is a decreasing function of θ. Both the supply side and the
adverse selection effect decrease the incentives of the LRs to carry cash and it is for this reason
that Md∗ is now a decreasing function of θ.
∗ changes with θ. As can be seen, P ∗ is a decreasing
Figure 5 illustrates how the price P2d
2d
b
function of θ. But note that the decline is more pronounced when θ < θ. The reason has
already been mentioned. As long as θ < b
θ an increase in θ has a double effect. A higher
θ worsens adverse selection concerns and thus the drop in prices. In addition, a higher θ
increases investment in the risky asset, which gets liquidated in states (ω 20 , ω 2L ). This supply
23
effect produces a further decline in prices that is absent when θ ≥ b
θ for then m∗d = 0 and there
can be no further investment in the risky asset. Notice that for θ > b
θ prices keep dropping
but a lower rate for now the supply is decreasing and thus the competition for the risky asset
amongst the LRs dampens the adverse selection effect on prices.
The pattern of returns is also revealing about the incentives of the LRs to carry outside
liquidity to the delayed-trading equilibrium. For θ < b
θ, R∗ (ω 20 , ω 2L ) is an increasing function
d
of θ: The expected payoff of the risky asset in the delayed-trading date
E [e
ρ3 |F] =
(1 − θ) ηρ
,
(1 − θη)
(17)
is a decreasing function of θ. But the price is dropping faster than returns on account both
of the adverse selection effect and the supply effect. This produces the increasing pattern in
returns. It is for this reason that the incentives of the LRs to carry outside liquidity are also
increasing in θ. Instead when θ > b
θ, the expected payoff is still decreasing but the decrease in
supply makes for a slow drop in prices as a function of θ, as we just saw, and thus the negative
slope of Rd∗ (ω 20 , ω 2L ) as a function of θ in this range.
In conclusion then, for θ ∈ [0, b
θ] the more severe the adverse selection problem, as
measured by θ, the higher the amount of outside liquidity brought to the market and the lower
the amount of inside liquidity carried by those holding the risky asset. This counterintuitive
result is due to the drop in prices, which makes the risky asset more attractive to the LRs in
states (ω 20 , ω 2L ). The larger the liquidity correction at date 2, the more attractive it is for LRs
to carry cash and trade opportunistically. Armed with these insights we turn to the question
of the Pareto ranking of the two equilibria.
III.D.4 Pareto ranking of the immediate and delayed-trading equilibria
Given these differences in ex-ante portfolio allocations an obvious question is whether
there a clear ranking of the two equilibria in terms of Pareto efficiency when they coexist?
Interestingly, the answer to this question is yes and, somewhat surprisingly, it is the delayed
trading equilibrium that Pareto dominates the immediate trading equilibrium. This is surprising, as delayed trade is hampered by the information asymmetry that arises in states (ω 20 , ω 2L ),
and therefore will take place at lower equilibrium prices.
Proposition 5. (Pareto ranking of equilibria.) Assume that A1-A4 hold and that δ is small
enough for all θ ∈ [0, θ] so that a delayed trading equilibrium always exists, then then
there exists a θ0 ∈ (0, θ] such that π ∗i ≤ π ∗d and Π∗i < Π∗d for all θ ∈ (0, θ0 ).
24
In our example θ0 = θ and though we have not been able to prove a tighter characterization of Proposition 5, we have been unable to find an example that meeting assumptions
A1-A4, features θ0 < θ. Thus in our example the delayed-trading equilibrium Pareto dominates
the immediate-trading equilibrium for all θ ∈ (0, θ]. This is illustrated in Figure 6, where the
expected profits of both the SRs and LRs are plotted for a particular range of θs.14
Figure 6 shows the expected profits of SRs and LRs as a function of θ for the delayed
trading equilibrium. The top panel shows the SRs’ expected profits. Notice that for θ ≤ b
θ=
.4196 the SRs are left at their reservation profits, which obtain if they were to be fully invested
in cash. Indeed, the SRs’ risky asset is a constant returns to scale technology and, a shown
in Proposition 4, in this range they are not fully invested in the risky asset. Figure 2 offered
a preview of this result. In that particular case θ = .35 < b
θ and thus the delayed-trading
equilibrium was located at the tangency point of the SR’s isoprofit line which corresponds
to its reservation value of π = 1 and the LR’s isoprofit line. The lower panel shows the LR’s
expected profit. The flat line corresponds to the LR’s expected profit in the immediate-trading
equilibrium, which is everywhere below the expected profit in the delayed-trading equilibrium.
What may be at first surprising is that the LR’s expected profits are, in this range, an increasing
function of θ: The higher the adverse selection the higher the LR’s expected profit. This again
is due to the fact that as we increase θ the asset is more likely to pay in date 2, when SRs care
the most for payoffs, and thus it becomes more attractive to them. This leads them to invest
more in the risky asset and carry less inside liquidity, which translates into more goods for the
LR in the event that the market opens at date 2. The liquidity premium associated with the
adverse selection problem combined with the increased supply of assets translates, as we saw
in Proposition 4, into an improvement of the investment opportunities available to the LRs at
the interim stage, which can only make them better off.
For θ > b
θ the SRs are fully invested in the risky asset and because of this fact they now
acquire some rents. Indeed for this range π ∗d > 1 and increasing with θ, whereas for the LRs
the expected profits are a decreasing function. Notice though that Π∗d > Π∗i throughout. A
particular example was depicted in Figure 3, where an example was shown where θ = .45 > b
θ.
As could be seen there, the delayed trading equilibrium was located strictly in the interior of
the lens formed by the two reservation isoprofit lines.
In Section IV below we explore the efficiency of equilibria more broadly and compare
equilibrium allocations to the allocations chosen by a planner at date 0 who seeks to maximize
a weighted sum of LR and SR payoffs subject to participation and incentive compatibility
14
The starting θ = .35 is simply chosen to show the figures in a convenient scale.
25
constraints. Consistent with our observation on Pareto ranked equilibria, the planner’s optimal
allocation is to enforce trade of the risky asset at date 2 in states (ω 20 , ω 2L ). The planner’s
overall objective is to maximize total surplus and total gains from trade. This requires both
limiting total cash reserves at date 0 and the trading of assets at date 2 in states (ω 20 , ω 2L ).
Interestingly, the planner’s optimal choice of cash holdings is generally strictly smaller than in
either of our two equilibria.
III.E Non-existence of the delayed-trading equilibrium and the value of commitment to trade
A maintained assumption throughout our analysis in section III.D is that the δ is small
enough so as not to compromise the existence of the delayed trading equilibrium. Here we
explore more deeply this assumption. Indeed, an important feature of our model is precisely
the fact that the SRs in state ω 2L may prefer to carry the asset to date 3 rather than trading
∗ at date 2.15 When this happens the delayed-trading exchange cannot be supported
it for P2d
as a competitive equilibrium for only lemons would appear in the market. SRs in state ω 2L
would have an incentive to retain the asset and carry it to date 3 whenever
Ped (ω 20 , ω 2L ) < δηρ,
(18)
where Ped (ω 20 , ω 2L ) is the candidate price for the risky asset at date 2 constructed as in
Proposition 2. In our example this occurs for the range economies for which
θ ∈ (.4628, .4834)
To illustrate graphically the welfare costs associated with this lack of existence, Figure 7 plots
the expected profits for the SRs and the LRs as a function of θ where we have selected the
Pareto superior equilibrium whenever there are two of them. There are three regions in the
plot. The first two correspond to the cases already discussed. In region A, θ ∈ (0, .4196)
the delayed trading equilibrium is Pareto superior and is such that mast
> 0. Region B,
d
θ ∈ [.4196, .4628), also features the delayed trading equilibrium as the Pareto superior one and
in that range of θs, m∗d = 0. Region C is one where, though assumptions A1-A4 are met, a
delayed-trading equilibrium cannot be supported and thus the immediate trading equilibrium
gets selected.
The dashed line in both panels of Figure 7 shows the additional expected profits that
would accrue to SRs and LRs if the former could commit ex-ante to liquidate their assets at the
15
It is easy to check that A2 implies that it is never optimal to retain the asset in an immediate trading
equilibrium constructed as above.
26
candidate price Ped (ω 20 , ω 2L ) in state ω 2L . In this case, the LRs anticipating that the pool of
assets supplied in states (ω 20 , ω 2L ) would also include assets of higher quality would be willing
to bring more outside liquidity than in the immediate trading equilibrium. As shown above,
this is always Pareto improving in our framework because it substitutes inside with outside
liquidity.
IV. LONG-TERM CONTRACTS FOR LIQUIDITY
So far we have only allowed SR and LR investors to trade assets for cash at dates 1
and 2, after they have each made their portfolio investment decisions at date 0. Given that
SR investors could reduce the lemons’ cost associated with outside liquidity by committing to
sell their long-term assets at date 2, the question naturally arises whether some form of long
term contract between an SR and LR at date 0 may improve on the outcome obtained in the
immediate and delayed-trading equilibria. We address this question in this section and allow
LRs to write long-term contracts with SRs at date 0, whereby an SR invests her endowment
in a fund managed by an LR and in return LR promises state-contingent payments to SR.
More precisely, the fund allocates the total endowment (1 + κ) of the two investors in
a portfolio that may comprise the LR long-run asset, the SR risky asset, and cash. The LR
manager in turn promises state contingent payments
{C1 (ω 1ρ ) , C1 (ω 1L ) , C2 (ω 2ρ ) , C2 (ω 20 ) , C2 (ω 2L ) , C3 (ω 3ρ ) , C3 (ω 30 )}
to SR for investing her unit of endowment in the fund. If the LR investor chooses to invest part
of the endowments in the SR risky asset then it is the LR investor who privately observes the
realization of the idiosyncratic shocks affecting the risky asset at dates 2 and 3. Accordingly,
an LR investor may also face incentive compatibility constraints, which limit the efficiency of
the long-term contract.16
Throughout this section we assume that δϕ0 (k) < 1, so that the LR investor would not
find it optimal to simply invest the whole endowment (1 + κ) in the long asset and repay the
SR at date 3. Incentive compatibility for the LR fund manager then requires first that:
C3 (ω 3ρ ) = C3 (ω 30 )
16
Note that if the SR investor can also observe the realization of idiosyncratic shocks then the asymmetric
information problem in the delayed-trading equilibrium would not be present, so that the long-term contract at
date 0 would clearly yields a superior outcome. The more consistent and interesting case, however, is when the
observation of idiosyncratic shocks is private information to the manager of the risky asset.
27
at date 3, for otherwise the LR simply announces the state which involves the lower payment
to SR, and second that at date 2,
C2 (ω 2L ) + C3 (ω 30 ) = C2 (ω 20 ) + C3 (ω 30 ) = C2 (ω 2ρ ) + C3 (ω 3ρ )
Otherwise, again, LR would always announce the state
(ω̂ 2 , ω̂ 3 ) ∈ {(ω 2L , ω 30 ), (ω 2L , ω 3ρ ), (ω 20 , ω 30 ), (ω 2ρ , ω 3ρ )}
which involves the lowest total payout C2 (ω̂ 2 ) + C3 (ω̂ 3 ).
The long term contract between LR and SR must also satisfy the SR and LR participation constraints. An individual SR investor has the choice of entering into a long-term
contract at date 0 with an LR or investing in a financial intermediary. The return offered by
financial intermediaries, in turn, depends on which equilibrium they expect. If they expect
the immediate-trading equilibrium then SR participation in a long-term requires that LR offer
at least the same payoff as financial intermediaries in that equilibrium. On the other hand,
if they expect the delayed-trading equilibrium the LR must offer at least the payoff in the
delayed-trading equilibrium.
As is easy to see, when SR investors expect the immediate-trading equilibrium, then
any pair of LR and SR investors are weakly better off writing a long-term contract at date 0.
Indeed, at worst the contract simply replicates the allocation under immediate trading. But,
the contract can also implement other allocations that are not feasible under the immediate
trading equilibrium, in particular by investing part of the SR endowment in the LR project
and by specifying payments to SR that (weakly) dominate the allocation under immediate
trading:
C1 (ω 1ρ ) ≥ m∗i + λ(1 − m∗i )ρ
C1 (ω 1L ) ≥ m∗i + (1 − m∗i )Pi∗ (ω 1L ).
It follows that:
Proposition 6. (Ranking of long-term contract and immediate-trading equilibrium) The
optimal long-term contract weakly (and sometimes strictly) dominates the equilibrium
allocation under immediate trading.
In contrast, when SR investors expect the delayed-trading equilibrium, then the longterm contract cannot always replicate the allocation under delayed trading. The reason is that
under delayed trading, SRs face different incentive constraints at date 2 from those faced by
LRs under the long-term contract.
28
Under delayed trading, the SR investor seeking to trade assets for cash at date 2 must
trade assets at the same price in states ω 20 and ω 2L to be induced to truthfully reveal these two
states to LRs. However, in state ω 2ρ there is no trade between SR and LR, and therefore also
no need to reveal that state truthfully to LR. In other words, no incentive constraint applies
in this state of nature.
Under the long-term contract, however, LR promises transfers to SR in four different
realizations of the idiosyncratic shocks {(ω 20 , ω 30 ); (ω 2ρ , ω 3ρ ); (ω 2L ; ω 30 ); (ω 2L ; ω 3ρ )} (see Figure X). As we have argued above, the total transfers in each of these four states must be the
same to satisfy incentive compatibility. Therefore, LR simply cannot replicate the allocation
under the delayed trading equilibrium with a suitable long-term contract. Given that the
delayed-trading equilibrium allocation is not in the feasible set for the long-term contract it is
not obvious a priori which allocation is superior. To be able to answer this question we must
first characterize the optimal long-term contract (when SR requires a payoff at least as high
as under the delayed trading equilibrium).
Solving the long-term contracting problem is a somewhat involved constrained optimization problem, as it involves two investment variables (α, M ) and seven state-contingent
transfers to SR. This problem can be simplified to some extent, as the next lemma establishes,
since the combination of all the incentive and feasibility constraints reduce the long-term contracting problem to the determination of optimal values for only: i) the amount α ∈ [0, 1]
invested in the risky SR project, ii) the amount M of cash held by the fund, and iii) payments
to SR in states ω 1ρ , ω 2ρ and ω 30 .
Lemma 7. Without loss of generality, any feasible, incentive-compatible long-term contract
between LR and SR takes the form:
C2 (ω)
C3 (ω)
ω 1ρ
M + αρ
C3 (ω 1ρ )
ω 2ρ
C2 (ω 2ρ )
C3 (ω 2ρ )
ω 20
M
C3 (ω 30 )
ω 2L , ω 30
M
C3 (ω 30 )
ω 2L , ω 3ρ
M
C3 (ω 3ρ )
Given that SR discounts date 3 consumption by δ it seems inefficient to offer any date 3
consumption to SR. Still, we cannot rule out that C3 (ω) > 0 for either ω ∈ {ω 1ρ , ω 2ρ , ω 30 , ω 3ρ }
since a date 3 transfer in one state may be required for LR to satisfy all incentive constraints
29
he faces. That is, to be able to credibly disclose that the realized state is ω 20 , for example,
LR may have to promise a high transfer C3 (ω 30 ) at date 3. Nevertheless, intuition suggests
that if δ is very small, λ sufficiently large, and the opportunity cost of holding cash for LR is
bounded, then the optimal contract ought to specify C3 (ω 1ρ ) = C3 (ω 2ρ ) = 0. This is indeed
the case as the following Proposition establishes.
Proposition 8. Suppose that δ is close to zero and that
η(1 − λ)ρ + ϕ(0) ≤ ϕ(κ),
(A5)
then the optimal long-term contract is such that C3 (ω 1ρ ) = C3 (ω 2ρ ) = 0.
With this characterization of the optimal long-term contract we are able to numerically
solve for the optimal contract and to compare LR payoffs under the contract to LR equilibrium
payoffs under the delayed trading equilibrium. We then show that for some parameter values,
the long-term contract is dominated by the delayed trading equilibrium outcome for high values
of θ.
The economic logic behind this result is that when θ is high then the SR risky asset
already matures most of the time at dates 1 or 2. The added value of additional liquidity
offered by LR through a long-term contract is then not that high. In addition, when θ is high
LR also faces high costs of meeting incentive constraints under the long-term contract. To be
able to credibly claim that the risky asset did not yield a return ρ at either dates 1 or 2, LR
must commit to wasteful date 3 payments C3 (ω 3ρ ) = C3 (ω 30 ) = αρ which SR does not value
much. The deadweight cost of these distortions exceeds the benefit of extra liquidity insurance,
which is why the delayed-trading equilibrium outcome is superior.
Example. In our example we keep θ as a free parameter and fix the other parameters to the
following values:
λ = .7
η = .4
ρ = 1.25
k = .12
δ = .1 and ϕ (x) = xγ
with γ = .19.
Note that all our assumptions are then met as long as θ ≤ .8148, which is the first value
of θ for which the assumption [λ + (1 − λ) η [θ + (1 − θ)δ]] ≤ 1 is violated. Accordingly
our plots below are restricted to the interval θ ∈ [0, .8148].
The payoffs of SR and LR under the long-term contract are given by respectively:
30
π ∗x = λ[M + αρ + δC3 (ω 1ρ )] +
(1 − λ)[θη(C2 (ω 2ρ ) + δC3 (ω 2ρ )) + (1 − θη)(M + δC3 (ω 30 ))]
and
Π∗x = M + ϕ[κ + (1 − α) − M ] +
λ[αρ − (C2 (ω 1ρ ) + C3 (ω 1ρ ))] +
(1 − λ)[ηαρ − (M + C3 (ω 30 ))].
In our numerical example we set π ∗x = π ∗d the SR payoff in the delayed trading equilibrium
to satisfy the SR participation constraint. Numerical computations show that for the
chosen parameter values the optimal long-term contract is such that C3 (ω 1ρ ) = C3 (ω 2ρ ) =
C3 (ω 30 ) = 0, and therefore that C2 (ω 2ρ ) = M .
Note that unlike in the previous example, a delayed trading equilibrium always exists
here. In the top panel of Figure 9 we graph the expected utility of the SR in the delayed
trading equilibrium as a function of θ whereas the bottom panel shows the expected
utility of the LR in the delayed trading equilibrium, Π∗d , as well as the LR’s expected
payoff under the long-term contract, Π∗x . For θ > θ̃ this payoff is less than what LR gets
in the delayed trading equilibrium. The bottom panel of Figure 10 shows that when θ
increases, the amount of cash carried by the LR to fulfill his commitments under the
long-term contract increases, making the contract less efficient, in sharp contrast with
the total amount of cash m∗d + Md∗ carried by both the LRs and SRs in the delayed
trading equilibrium, shown in the top panel of the same figure. This increase in cash
follows because the expected payoff of SR in the delayed trading equilibrium increases
with θ. Incentive constraints limit the difference in payments in states ω 2ρ and ω 20 , and
since payments at date 3 are very inefficient, the contract specifies higher payments at
date 2, which requires carrying more cash.
V. REGULATORY IMPLICATIONS
V.A Minimum reserve requirements and the scope of regulation
The source of liquidity demand is a potential maturity mismatch problem for SR investors: they prefer to consume at dates 1 and 2, but the risky asset may only pay at date 3.
31
Financial intermediaries that engage in maturity transformation, such as banks and insurance
companies, are heavily regulated in most jurisdictions in an effort to protect claimants from
potential losses on the asset side of these institutions’ balance sheets. One common regulatory
response is the imposition of minimum reserve or liquidity requirements that compel banks
to maintain a certain percentage of their assets in “cash” or other liquid securities such as
treasury bills.
To analyze the effect of these minimum reserve requirements it is useful to turn to the
example illustrated in Figure 3, where the efficient delayed trading equilibrium is such that SRs
are fully invested in the risky asset, m∗d = 0. Clearly any reserve requirement in this situation
risks undermining the delayed-trading equilibrium and to leave as the only equilibrium the
immediate-trading equilibrium. The general equilibrium effects are such that when SRs are
required to carry a minimum amount of cash the returns of cash holdings in the unregulated
sector go down, as SRs now invest less and sell fewer risky assets. This in turn makes the risky
assets less attractive to SRs. Also, the resulting equilibrium outcome could be one where the
minimum cash position exceeds the regulatory requirement in equilibrium.
V.B Value-at-Risk (VAR) regulation and mandated asset sales
As we have shown, SR investors’s option to postpone consumption until date 3 may
∗ ≤ δηρ. A regulaundermine existence of the efficient delayed-trading equilibrium when P2d
tory intervention mandating SRs to liquidate assets at date 2 could thus conceivably improve
efficiency by creating an environment that helps support a delayed-trading equilibrium. A
VAR regulation requiring financial institutions to sell assets when too much value is at risk
may be seen as achieving such an intervention.17 Although a surprise increase in asset sales at
date 2 only results in lower prices, the anticipation of such sales would encourage LRs to hold
more cash and SRs less, thus increasing ex-ante efficiency. Once again the benefits of such an
intervention obtain through a general equilibrium effect. In the presence of a VAR regulation
the returns of carrying outside liquidity increase for the LRs for they now know that the all
SRs will be forced to liquidate assets, the lemons as well as the good assets.
V.C Public and private provision of liquidity
V.C.1 Inefficient private supply of outside liquidity
So far we have assumed that outside liquidity is supplied competitively, so that LRs
17
However, VAR regulation that results in financial institutions holding on to their assets and raising more
regulatory capital would be counterproductive.
32
∗ . A
do not take into account the effect that their choices have on the equilibrium price P2d
monopoly LR, on the other hand, would internalize the effect of its supply of liquidity on price.
The obvious question then arises whether a monopoly LR might be more efficient in situations
where the competitive delayed-trading equilibrium fails to exist?
When θ < b
θ, SRs carry a strictly positive amount of inside liquidity m∗d > 0 and make
zero profits. All the surplus goes to the LRs, who cannot obtain higher profits by changing
their holdings of liquidity. It follows then that in this range the competitive and monopoly
solutions are identical. In contrast, when θ ≥ b
θ, the level of inside liquidity in the competitive
equilibrium is m∗d = 0, LRs compete for a fixed supply of the risky asset, and SRs obtain some
of the surplus from trade. In this situation, a monopoly LR would gain by restricting its supply
∗ . This can be seen in Figure 8, where, in
of outside liquidity and thereby raising the price P2d
the top panel, the profits of the monopolist are plotted together with the competitive ones and
in the bottom panel the prices in states (ω 20 , ω 2L ) are plotted against θ.
Notice first that in region A, which corresponds to the case θ < b
θ, the prices and profits
in a monopoly are identical to those under perfect competition. In region B, SRs set the level
of inside liquidity to m∗d = 0 and the monopoly LR restricts the supply of outside liquidity so
as to capture fully all the gains from trade. This explains why the price of the risky asset in
states (ω 20 , ω 2L ) under a monopoly is below the competitive price.
But once θ exceeds a certain threshold the competitive delayed-trading equilibrium no
longer exists. The monopoly LR in this case has to set the price for the risky asset to δηρ
to guarantee a profitable trade at date 2. In this parameter region, region C in Figure 8,
a monopoly LR will indeed improve efficiency. By carrying and supplying enough outside
liquidity the monopolist elicits the supply of risky assets by SRs in state ω 2L , thus avoiding
the break down of the delayed exchange. Notice that as shown in the top panel of Figure 8,
the monopoly’s profits are, in this region, above those that obtain in the immediate-trading
equilibrium, which is the only one that exists with competitive LRs.18
V.C.1 Public liquidity as complementary to private outside liquidity
In the more plausible situation where outside liquidity is supplied competitively, the
price support function played by a monopoly LR has to be taken up the public sector. Notice
that in this case the provision of public liquidity is complementary to the private provision
of liquidity. Indeed, when the public provision of liquidity increases the price of the risky asset
18
It is worth emphasizing than in this region SR profits are such that π > 1. The reason is that the monopolist
has to “leave some rents” to the SRs precisely to elicit the supply of the assets in state ω 2L .
33
up to δηρ in region C, LRs have an incentive to also carry outside liquidity and purchase the
risky assets supplied in states (ω 20 , ω 2L ). In contrast, if the public sector supplies liquidity in
regions A or B it will crowd out liquidity provided by LRs. In this parameter region public
and private liquidity are substitutes rather than complements. It follows then that whether
the public provision of liquidity is beneficial or not depends critically on the degree of adverse
selection in the market: If it is high enough to prevent the delayed exchange from occurring
then there is an efficiency improving role for public liquidity which encourages in turn the
private supply of outside liquidity.
VI. CONCLUSIONS
This paper is concerned with three questions. First, what determines the distribution
of liquidity across market participants? Second, is this distribution (constrained) efficient?
Finally, if it is not efficient, what are the regulatory remedies that can restore efficiency?
A novel dimension of our model is the cross sectional supply of liquidity, which seems to
be a core feature of modern financial markets, where different actors outside the regulated
financial intermediary sectors that stand ready to absorb asset sales by distressed financial
intermediaries.19 The incentives of the different parties to carry liquidity in our model are
driven by their different opportunity costs and different investment horizons. An important
question we address is whether a competitive price mechanism would elicit the optimal crosssectional cash reserve decisions by all the different actors.
A second element in the model that departs from the existing literature is the endogenous
timing of asset sales and the deterioration of adverse selection problems over time. Financial
intermediaries face the choice of raising liquidity early, or in anticipation of a crisis, before
adverse selection problems set in, or in the midst of a crisis at more depressed prices. The
benefit of delaying asset sales and attempting to ride through the crisis is that the intermediary
may be able to entirely avoid any sale of assets at distressed prices should the crisis be short
and mild. We show that when the adverse selection problem is not too severe there are multiple
equilibria, an immediate-trading and a delayed-trading equilibrium. In the first equilibrium,
19
For instance, a recent article in the Wall Street Journal of Friday May 9th 2008 by Lingling WeI and
Jennifer S. Forsyht emphasized that the discounts on commercial real estate debt are less pronounced than
in the previous real-estate collapse of the early 1990s. As the authors point out “[t]oday there are at least
55 active or planned commercial real-estate debt funds seeking to raise $33.8 billion, according to Real Estate
Alert, a trade publication. And many have begun to do deals.” In the recent period of distress that started
in the summer of 2007, the role of sovereign funds has been notorious and major source of recapitalization for
institutions such as Citi.
34
intermediaries liquidate their positions in exchange for cash early in the liquidity crisis. In the
second equilibrium, liquidation takes place late in the liquidity event and in the presence of
adverse selection problems.
We show that surprisingly the latter equilibrium Pareto-dominates the former because
it saves on cash reserves, which are costly to carry. However, the delayed-trading equilibrium
does not exist when the adverse selection problem is severe enough. The reason is that in
this case prices are so depressed as to make it profitable for the agents holding good assets to
carry them to maturity even when it is very costly to do so. We show that if they were able
to do so, intermediaries would be better off committing ex-ante to liquidating their assets at
these depressed prices in the distressed states. We also show, perhaps more surprisingly, that
a monopoly supplier of liquidity may be able to improve welfare.
Thus, we argue that the role of the public sector as a provider of liquidity has to be
understood in the context of a competitive provision of liquidity by the private sector. We
show, in particular that public provision of liquidity can act as a complement for private
liquidity in situations where lemon’s problems are so severe that the market would break down
without any public price support. An important assumption in our analysis, however, is that
the public liquidity provider is able to determine which state of nature the economy is in (ω 1L
or (ω 20 , ω 2L )) and to adequately adapt its response to circumstances. An important remaining
task is thus to analyze the benefits of public policy in our model under the assumption that
the public agency may be ignorant about the true state of nature in which it is intervening.
Another central theme in our analysis is the particular timing of the liquidity crisis that
we propose. Liquidity crisis have, in our opinion, always real origins, small as they may be. In
our framework the onset of the liquidity event starts with a real deterioration of the quality of
the risky asset held by financial intermediaries. It is only later that the adverse selection sets
in. Our purpose here was to capture the fact that the market may realize slowly the extent to
which many participants are affected by the deterioration in asset values. Financial institutions
here face a choice of whether to liquidate early or ride out the crisis in the hope that the asset
may ultimately pay off. This trade-off is unrelated to the incentives that may force institutions
to liquidate at particular times due to accounting and credit quality restrictions in the assets
they can hold that have featured more prominently in the literature. Understanding the effect
that these restrictions have on the portfolio decisions of the different intermediaries remains
an important question to explore in future research.
Finally, in our model long run investors are those with sufficient knowledge to be able to
value and absorb the risky assets for sale by financial intermediaries. Only their capital and
35
liquid reserves matter for equilibrium pricing to the extent that they are the only participants
with the knowledge to perform an adequate valuation. Other, less knowledge intensive, capital
will only step in at steeper discounts for which there may be no market. Our current work
focuses precisely on understanding how different knowledge-capital gets “earmarked” to specific
markets. What arises is a theory of market segmentation and contagion that, we believe may
shed light on the behavior of financial markets in states of crisis.
36
REFERENCES
Acharya, Viral (2001) “A Theory of Systemic Risk and Design of Prudential Bank Regulation” manuscript,
New York University and London Business School.
Acharya, Viral and Tanju Yorulmazer (2005) “Cash-in-the-market Pricing and Optimal Bank Bailout
Policy,” manuscript, CEPR.
Aghion, Philippe, Patrick Bolton, and Mathias Dewatripont (2000) “Contagious Bank Failures in a
Free Banking System” European Economic Review, May.
Allen, Franklin and Douglas Gale (1998) “Optimal Financial Crisis,” Journal of Finance, 53, 1245-1284.
Bhattacharya, Sudipto and Douglas Gale (1986) “Preference Shocks, Liquidity, and Central Bank
Policy,” manuscript, CARESS Working Paper #86-01, University of Pennsylvania.
Brunnermeier, Markus and Lasse Pedersen (2008) “Market Liquidity and Funding Liquidity,” Review
of Financial Studies, forthcoming.
Bryant, John (1980) “A Model of Reserves, Bank Runs, and Deposit Insurance” Journal of Banking and
Finance, 3, 335-344.
Diamond, Douglas W. and Philip H. Dybvig (1983) “Bank Runs, Deposit Insurance, and Liquidity”
Journal of Political Economy, vol. 91 no. 3, 401-419.
Diamond, Douglas W. (1997) “Liquidity, Banks, and Markets” Journal of Political Economy, vol. 105 no.
5, 928-956.
Diamond, Douglas W. and Raghuram Rajan (2005) “Liquidity Shortage and Banking Crisis,” Journal
of Finance, April.
Dow, James and Gary Gorton (1993), “Arbitrage Chains”, NBER Working Paper No. W4314. Available
at SSRN: http://ssrn.com/abstract=478730
Fecht, Falko (2004) “On the Stability of Different Financial Systems,” Journal of the European Economic
Association, vol. 2 no. 6, 969-1014.
Freixas, Xavier, Bruno Parigi and Jean-Charles Rochet (2000) “Systemic Risk, Interbank Relations,
and Liquidity Provision by the Central Bank” Journal of Money, Credit and Banking, vol. 32 (2),
611-638.
Gorton, Gary and Lixin Huang (2004) “Liquidity, Efficiency, and Bank Bailouts,” American Economic
Review, vol. 94 no. 3, 455-483.
Gromb, Dennis and Dimitri Vayanos (2002) “Financially Constrained Arbitrage and the Cross-section
of Market Liquidity,” manuscript, London Business School.
Holmstron, Bengt and Jean Tirole (1998) “Private and Public Supply of Liquidity” Journal of Political
Economy, vol. 106 no. 1, 1-40.
Jacklin, Charles J. (1987) “Demand Deposits, Trading Restrictions, and Risk Sharing.” In E.C. Prescott,
and N. Wallace (ed.), Contractual Arrangements for Intertemporal Trade, University of Minnesota Press.
Kondor, Peter (2008) “Risk in Dynamic Arbitrage: The Price Effects of Convergence,” Journal of Finance,
forthcoming.
37
Kyle, Peter and Wei Xiong (2001) “Contagion as a Wealth Effect,” Journal of Finance, vol. 56, 14011440.
Parlour, Christine and Guillaume Plantin (2008) “Loan Sales and Relationship Banking,” Journal of
Finance, forthcoming.
Xiong, Wei (2001) “Convergence Trading with Wealth Effects,” Journal of Financial Economics, vol. 62,
247-292.
38
APPENDIX
Proof of Proposition 1. The first order condition of the LR is
λ + (1 − λ)
ηρ
≤ ϕ0 (κ − M ) .
Pi (ω 1L )
(19)
First we establish that it is not possible to support an equilibrium with Mi∗ = 0 and m∗i = 1. Indeed if
m∗i = 1 it has to be the case that the price in state ω 1L is such that
∗
≤
P1i
1 − λρ
1−λ
but by assumption A3 this implies
λ + (1 − λ)
ηρ
> ϕ0 (κ) ,
Pi (ω 1L )
(20)
and thus Mi∗ > 0 a contradiction.
Having ruled the no trade immediate trading equilibrium we proceed next as follows. Start by solving
the following equation in Pi (ω 1L )
λ + (1 − λ)
ηρ
= ϕ0 (κ − Pi (ω 1L )) ,
Pi (ω 1L )
and define
P =
1 − λρ
,
1−λ
(21)
(22)
as positive number by assumption A2.
∗
• Case 1: Assume first that Pi (ω 1L ) ≥ P , then set P1i
= Mi∗ = Pi (ω 1L ) and m∗i = 0, which meets
the first order condition of the SRs as can be checked by inspection of expression (2).
∗
• Case 2: Assume next that Pi (ω 1L ) < P , then set P1i
= P and Mi∗ to be the solution to
λ + (1 − λ)
ηρ
≤ ϕ0 (κ − Mi∗ ) ,
P
(23)
which by assumption A3 is such that Mi∗ > 0 and clearly it has to be such that Mi∗ < P . Because,
given these prices, the SRs are indifferent on the level of cash carried set m∗i so that
∗
P1i
=
1 − λρ
Mi∗
=
1−λ
1 − m∗i
(24)
As for prices in states (ω 20 , ω 2L ) they have to be such that both the SRs and the LRs prefer to trade at
ω 1L . For this set

ff
1 − ρ [λ + (1 − λ) θη]
Pi∗ (ω 20 , ω 2L ) < min δηρ,
.
(1 − λ) (1 − θη)
(25)
Given this price the LR investors expect only lemons (assets with zero payoff) in the market at ω 20 , ω 2L
and thus the demand is equal to zero Q∗ (ω 20 , ω 2L ) = 0. As for the SRs notice that if they wait to
liquidate at ω 20 , ω 2L they obtain
θηρ + (1 − θη) Pi∗ (ω 20 , ω 2L ) < θηρ +
1 − ρ [λ + (1 − λ)θη]
∗
= P1i
,
1−λ
and thus SRs set q ∗ (ω 20 , ω 2L ) = 0 and q ∗ (ω 1L ) = 1 − m∗i = Q∗ (ω 1L ).
39
2
Proof of Proposition 2. First notice that since ϕ0 (κ) > 1 in any delayed trading equilibrium there must
be cash-in-the-market pricing thus
∗
Md∗ = P2d
(1 − θη) (1 − m)
(26)
Define
(1 − θ) ηρ
= ϕ0 [κ − (1 − θη) Pd (ω 20 , ω 2L )]
(1 − θη) Pd (ω 20 , ω 2L )
This equation always a unique solution which in addition satisfies
„
«
κ
Pd (ω 20 , ω 2L ) ∈ 0,
.
1 − θη
λ + (1 − λ)
(27)
(28)
There are two cases to consider:
• Case 1: Pd (ω 20 , ω 2L ) is such that
1 − ρ [λ + (1 − λ) θη]
= P.
(1 − λ) (1 − θη)
Pd (ω 20 , ω 2L ) <
(29)
In this case set
∗
P2d
= P,
(30)
and set Md∗ to be the solution of
λ + (1 − λ)
(1 − θ) ηρ
0
∗
∗ = ϕ (κ − Md ) ,
(1 − θη) P2d
(31)
which from the strict concavity of ϕ(·) is
∗
Md∗ < (1 − θη)P2d
.
Choose m∗d such that
∗
P2d
=
Md∗
(1 − θη) (1 − m∗d )
(32)
(33)
∗
Notice that because P2d
= P the SRs are indifferent in the level of cash held. Both types of traders
would prefer to wait to trade at date 2 provided that Pd∗ (ω 1L ) is in the interval
»
–
∗
(1 − θη) P2d
∗
, θηρ + (1 − θη) P2d
,
1−θ
which is non empty if and only if
∗
P2d
≤
(1 − θ) ηρ
= P.
1 − θη
(34)
(35)
Clearly, given assumption A1, specifically the fact that ϕ0 (κ) > 1, and equation (31), equation (35)
is trivially met. Clearly, given assumption A1, specifically the fact that ϕ0 (κ) > 1, and equation
(31), equation (35) is trivially met.
∗
Notice that P2d
is independent of δ and for δ ≤ δ, where
δ=
∗
P2d
,
ηρ
∗
the SR (weakly) prefers to trade at date 2 for a price P2d
than carrying the asset to date 3.
40
(36)
• Case 2: Pd (ω 20 , ω 2L ) ≥ P , then choose
∗
P2d
= Pd (ω 20 , ω 2L )
∗
Md∗ = P2d
(1 − θη)
and
m∗d = 0.
(37)
Except for establishing inequality (35), the remainder of the proof follows as in the previous case.
∗
To establish that P2d
meets (35) it is enough to substitute P in (35) and appeal to assumption A2.
2
Before proving Propositions 3, 4, and 5, it is useful to establish the following
Result. Assume A1-A4 hold. Then the immediate trading equilibrium is such that m∗i ∈ (0, 1).
Proof. By the SR’s first order condition if the price at date 1 is given by
∗
P1i
=
1 − λρ
1−λ
then the SR investor is indifferent about the cash position carried. Let Mi∗ be the solution to
λ + (1 − λ)2
ηρ
= ϕ0 (κ − Md∗ ) ,
1 − λρ
which by assumption A3 exists and is unique. By assumption A4,
1 − λρ
> κ > Mi∗ .
1−λ
Then set m∗i ∈ (0, 1) so that
1 − λρ
Mi∗
=
.
1−λ
1 − m∗i
The construction now of the immediate trading equilibrium follows as in the proof of Proposition 1.
2
We prove Proposition 4 first. The proof of Proposition 3 following trivially after that.
Proof of Proposition 4 First notice that by the result above, the immediate trading equilibrium is such
that m∗i > 0 (and, obviously, Mi∗ > 0). Thus because the delayed trading equilibrium specializes to the
immediate trading equilibrium when θ = 0, it follows that there exists a neighborhood (0, θ̃) such that
m∗d > 0. Then from the LR’s and SR’s first order conditions, combined with cash in the market pricing,
Md∗ and m∗d are fully determined by
ψ (M )
ψ
(m)
=
=
λ + (1 − λ) Rd∗ (θ) − ϕ0 (κ − Md∗ ) = 0
(1 −
m∗d ) (1
− ρ (λ + (1−) θη)) − (1 −
(38)
λ) Md∗
=0
(39)
Expression (38) is the LR’s first order condition. Expression (39) is the SR’s first order condition combined
with the cash-in-the-market pricing equation. These two equations determine Md∗ and m∗d . In the above
expression
Rd∗ =
(1 − θ) ηρ
∗ ,
(1 − θη) P2d
∗
where P2d
is given by P (see expression (29). Then basic algebra shows that
∗
Rd,θ
=
∂Rd∗
∝ ρ [λ + (1 − λ) η] − 1 > 0,
∂θ
by assumption (A1).
41
(40)
(M )
∂x ψ =
ψM
(m)
ψM
)
ψ (M
m
!
(M )
and
ψ (m)
m
∂θ ψ =
ψθ
!
(m)
ψθ
,
(41)
where
ψM
(M )
=
ϕ00 (κ − Md∗ ) < 0
)
ψ (M
m
=
0
ψ (m)
m
=
− [1 − ρ (λ + (1 − λ)θη)] < 0
(m)
ψM
=
−(1 − λ)
(M )
=
∗
(1 − λ)Rd,θ
>0
(m)
=
−(1 − m∗d )(1 − λ)ηρ < 0,
ψθ
ψθ
First,
|∂x ψ| = − [1 − ρ (λ + (1 − λ)θη)] ϕ00 (κ − Md∗ ) > 0
Second, by an application of the implicit function theorem
∗
=
Md,θ
∂Md∗
= −[ 1
∂θ
−1
0 ] (∂x ψ) ∂θ ψ
m∗d,θ =
and
∂m∗d
= −[ 0
∂θ
−1
1 ] (∂x ψ) ∂θ ψ.
After some algebra:
m∗d,θ
=
h
i
(m)
(M )
∗
−|∂x ψ|−1 −ψ M (1 − λ) Rd,θ
− ψ M (1 − m) (1 − λ) ηρ
ˆ
˜
∗
−|∂x ψ|−1 (1 − λ)2 Rd,θ
− ϕ00 (κ − Md∗ ) (1 − m∗d ) (1 − λ) ηρ
<
0
=
(42)
and
∗
Md,θ
=
h
i
(M )
(M ) (m)
−|∂x ψ| ψ (m)
− ψm
ψθ
m ψθ
=
−|∂x ψ|ψ (m)
m ψθ
>
0
(M )
Because m∗d is strictly decreasing in θ if m∗d = 0 for some b
θ, then m∗d = 0 for all θ ≥ b
θ. For θ ≥ b
θ the
LR’s first order condition is given by
λ + (1 − λ)
(1 − θ) ηρ
= ϕ0 (κ − Md∗ ) ,
Md∗
where we have made use of the fact that cash-in-the-market pricing obtains and m∗d = 0. Then a basic
∗
application of the implicit function theorem shows that Md,θ
< 0 for θ > b
θ. As for the behavior of
expected returns when θ > b
θ, notice that the LR’s first order condition is written as
λ + (1 − λ) Rd∗ = ϕ0 (κ − Md∗ ) ,
42
a
∗
and thus given that Md,θst < 0 for θ > b
θ, it follows that Rd,θ
< 0 for that range.
We turn now to the properties of the aggregate supply of the risky asset at date 2 in the delayed trading
equilibrium s∗d . Using (42),
s∗d,θ
=
∗
|∂x ψ|−1 (1 − λ)2 Rd,θ
(1 − θη)
−
|∂x ψ|−1 ϕ00 (κ − m∗d ) (1 − m∗d ) (1 − λ) ηρ (1 − θη) − η (1 − m∗d )
(43)
Tedious algebra shows that (43) is equal to
»
(1 −
m∗d ) η
–
ρ−1
,
1 − ρ (λ + (1 − λ)θη)
which is positive by assumption A2. This completes the proof of Proposition 3.
m∗i
2
m∗i
m∗d
m∗d
(θ = 0) and
follows immediately from the fact that
=
>
∗
∗
b
b
Proposition 3. Clearly for θ ≤ θ where θ was defined in the proof of Proposition 3, Mi < Md . For θ > b
θ
Proof of Proposition 4. That
Md∗ is a decreasing function of θ and thus, by continuity there exists a (unique) θ0 , possibly higher than
θ, for which Md∗ (θ0 ) = Mi∗ ; for any θ < θ 0 , Mi∗ < Md∗ .
2
Proof of Proposition 5. Under assumption A4, m∗i > 0 and thus π ∗i = 1 ≤ π ∗d . As for the expected profits
of the LR investors, firs notice that
∂Π∗d
∗
= Π∗d,θ = (1 − λ) Rd,θ
Md∗ .
∂θ
Given that Π∗i = Π∗d (θ = 0) and the characterization of expected returns in Proposition 3 the result
follows immediately.
2
Proof of Lemma 7. Given a choice M of cash carried by LR and α invested in the SR risky project
(0 ≤ α ≤ 1 ), the feasibility constraints on transfers to SR are given by:
C1 (ω 1ρ )
≤
αρ + M,
C1 (ω 1ρ ) + C3 (ω 1ρ )
≤
αρ + M + ϕ [κ + (1 − α) − M ] ,
C1 (ω 1L ) + C2 (ω 20 )
≤
M,
C1 (ω 1L ) + C2 (ω 2L )
≤
M,
C1 (ω 1L ) + C2 (ω 2ρ )
≤
αρ + M,
C1 (ω 1L ) + C2 (ω 2ρ ) + C3 (ω 2ρ )
≤
αρ + M + ϕ [κ + (1 − α) − M ] ,
C1 (ω 1L ) + C2 (ω 20 ) + C3 (ω 20 )
≤
M + ϕ [κ + (1 − α) − M ] ,
C1 (ω 1L ) + C2 (ω 2L ) + C3 (ω 30 )
≤
M + ϕ [κ + (1 − α) − M ] .
Consider next the following observations concerning equilibrium contracts:
I. State ω 1ρ is observable and since there is no discounting between periods 1 and 2 we may assume without
any loss of generality that C1 (ω 1ρ ) = C1 (ω 1L ) = 0.
II. If C3 (ω 1ρ ) > 0 then C2 (ω 1ρ ) = αρ + M . For if C2 (ω 1ρ ) < αρ + M , both agents can be made better off
by increasing C2 (ω 1ρ ) and decreasing C3 (ω 1ρ ).
43
III. Incentive compatibility requires that C3 (w30 ) = C3 (w3ρ ). Hence any feasible and incentive compatible
payment in histories that follow from ω 2L is also feasible in histories that follow ω 20 . Incentive compatibility also requires that
C2 (ω 2L ) + C3 (ω 30 ) = C2 (ω 20 ) + C3 (ω 20 ) .
Therefore any payment prescribed for the histories starting at ω 2L must also be prescribed for histories
starting at ω 20 :
C2 (ω 2L ) = C 2 (ω 20 ),
and
C3 (ω 30 ) = C 3 (ω 20 ).
IV. If C3 (ω 30 ) > 0 then C2 (ω 2L ) = M. For if C2 (ω 2L ) < M SR can be made better off, while keeping LR
indifferent, by increasing the payment at date 2 and decreasing by the same amount the payments in states
ω 30 and ω 3ρ at date 3. The same reason, together with observation 3, implies that if C3 (ω 20 ) > 0 then
C2 (ω 20 ) = M . We can also use the same reasoning to show that if C3 (ω 2ρ ) > 0 then C2 (ω 2ρ ) = M + αρ.
V. Since (λ + (1 − λ)) ρ > 1
and ϕ0 (κ) > 1,
if cash is carried by the LR it must be distributed in
some state (at either dates 1 or 2). Hence either C2 (ω 1ρ ) = M + αρ, or C2 (ω 2ρ ) = M + αρ or
C2 (ω 20 ) = C2 (ω 2L ) = M . Note furthermore from observation 4 and incentive compatibility that we
must have C2 (ω 2ρ ) > 0 and C2 (ω 20 ) > 0 unless SR consumption is zero in all histories starting at ω 1L .
However, in the latter case, because of discounting and δφ0 (k) < 1 the ex-ante contract is dominated by
autarky. Hence we may assume that C2 (ω 2ρ ) > 0 and C2 (ω 20 ) > 0 . In an analogous fashion we can
establish that C2 (ω 1ρ ) > 0.
VI. Suppose, that C2 (ω 1ρ ) ≤ M + αρ − µ for some µ > 0, and let γ > 0 be small enough that γ <
λ
γ 1−λ
µ
2
and
< min{C2 (ω 2ρ ); C2 (ω 20 )}. Consider the payment
Ĉ2 (ω 1ρ ) = C2 (ω 1ρ ) + γ
λ
. This new contract, leaves SR
and lower date 2 payments for all realizations following ω 1L by γ 1−λ
indifferent and economizes in cash. This cash can be invested in the LR project, which has a marginal
product above one, and yield extra utility for LR at date 3. Hence the initial contract cannot be optimal.
VII. Suppose that C2 (ω 2L ) < M , then from observation 4, C3 (ω 30 ) = 0. Hence C2 (ω 20 ) < M and C2 (ω 2ρ ) <
M + αρ. Using the same logic as in observation 6 we may then show that this contract is not optimal.
VIII. Incentive compatibility requires that
C 2 (ω 2ρ ) + C 3 (ω 2ρ ) = M + C 3 (ω 30 ).
Since C2 (ω 2ρ ) = M satisfies the LR budget constraint, it follows that
C3 (ω 2ρ ) ≤ C3 (ω 30 ).
This concludes the proof.
2
Proof of Proposition 8. Under assumption (A5) LR’s opportunity cost of holding cash, ϕ0 (κ+(1 − α) − M ),
is bounded. To see this, note first from lemma 7 that LR must pay SR at least M following the realization
of state ω 1L . LR’s date 0 expected payoff therefore cannot exceed:
η(1 − λ)αρ + ϕ(κ + (1 − α) − M ).
44
Since participation by LR requires that
η(1 − λ)αρ + ϕ(κ + (1 − α) − M ) ≥ ϕ(κ),
we must have κ + (1 − α) − M > 0, by assumption A5. It follows that
ϕ0 (κ + (1 − α) − M ) < B, for some B > 0.
Now, suppose by contradiction that C3 (ω 1ρ ) ≥ ² > 0. Then lowering C3 (ω 1ρ ) by ² and increasing M by
δλ² keeps SR indifferent, but makes LR strictly better off if Bδ < 1. Similarly, if min{C3 (ω 2ρ ); C3 (ω 30 )} =
C3 (ω 2ρ ) ≥ ², a decrease of C3 (ω 30 ) and C3 (ω 2ρ ) by ² and an increase of M by (1 − λ)δ², again keeps
SR indifferent but makes LR better off (provided that Bδ < 1).
45
2
λ
©©
* ω 1ρ
©
©
©©
©
©©
ω0 ©
H
H
HH
*
η ©©
©
©
©
* HH
θ ©©
©
HH
©
HH
H
1−λ H
HH
j
ω 2ρ
¶³
j ω 20
H
1−η
ω 1L ©
H
H
HH
H
1−θ
HH
HH
j
H
*
©©
η ©©
©©
©©
ω 3ρ
©
ω 2L ©
HH
µ´
HH
HH
1 − η HH
HH
j
ω 30
t=0
t=1
t=2
t=3
Figure 1. The risky asset. There are four dates. Investment in the risky asset occurs at date0. At date 1
there is an aggregate shock. Specifically there are two possible aggregate states, ω 1ρ , which occurs with
probability λ, and ω 1L , which occurs with probability 1 − λ. In ω 1ρ the risky asset matures at date 1
and yields a cash dividend ρ. ω 1L is the state when the long duration asset matures later than date 1
(either at dates 2 or 3). At date 2, there are three idiosyncratic states of nature, ω 2ρ , which occurs with
probability θη, ω 20 , which occurs with probability θ (1 − η),and ω 2L , which occurs with probability 1 − θ.
ω 2ρ is the state when the asset matures at date 2 and yields dividend ρ. Thus the probabilities also
denote the mass of SRs which are in the corresponding states of nature. ω 20 is the state when the asset
matures at date 2 but yields no dividends. In ω 2L the risky asset is known to mature at date 3. Finally
in date 3 there are again two states, ω 3ρ , which occurs with probability η, and ω 30 , which occurs with
probability 1 − η. In state ω 3ρ the asset matures at date 3 and yields dividend ρ and in state ω 30 the
asset matures at date 3 and yields zero dividends. The information set of the LRs at date 2 is given by
{ω 20 , ω 2L }, which generates the adverse selection problem that is key in the analysis. At date 1, agents
are symmetrically informed.
46
Immediate versus delayed trading equilibrium: θ=.35
1
(M*,m*)
i
0.9
i
0.8
0.7
m
0.6
0.5
0.4
*
*
(Md,md)
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
0.1
M
Figure 2. Immediate and delayed-exchange equilibria in the Example for the case θ = .35. The graph
represents cash holdings, with the cash holdings of the LRs in the x-axis and the cash holdings of the SRs
in the y-axis. The dashed curves represent isoprofit lines for the LR and the straight continuous lines
represent the SR’s isoprofit lines, for both when the exchange occurs in state ω 1L and in states (ω 20 , ω 2L ).
The isoprofit lines for the SR correspond to its reservation profits π ∗i = π ∗d = 1. The immediate and
delayed-trading equilibrium cash holdings are marked (Mi∗ , m∗i ) and (Md∗ , m∗d ), respectively.
47
Immediate versus delayed trading equilibrium: θ=.45
(M*i,m*i)
0.9
0.8
0.7
0.6
m
0.5
0.4
0.3
*
*
(Md,md)
0.2
IPSR
0.1
0
−0.1
0
0.02
0.04
0.06
0.08
0.1
M
Figure 3. Immediate and delayed-exchange equilibria in the example when θ = .45. The graph represents
cash holdings, with the cash holdings of the LRs in the x-axis and the cash holdings of the SRs in the
y-axis. The dashed curves represent isoprofit lines for the LR and the straight continuous lines represent
the SR’s isoprofit lines, for both when the exchange occurs in state ω 1L and in states (ω 20 , ω 2L ). As
opposed to the case in Figure 2 now the delayed-treading equilibrium, marked (Md∗ , m∗d ), has the SRs
commanding strictly positive profits, π ∗d > 1. The line marked IPSR denotes the SR’s reservation isoprofit
line in states (ω 20 , ω 2L ).
48
Cash position of the SRs in the delayed trading equilibrium as a function of θ
m*d(θ)
0.6
0.4
0.2
0
0.35
0.45
θ
Cash position of the LRs in the delayed trading equilibrium as a function of θ
0.08
M*d(θ)
0.4
0.07
0.06
0.05
0.35
0.4
θ
0.45
Figure 4. Cash holdings as a function of θ for the Example. Panel A represents the SR’s cash holdings in
the delayed-trading equilibrium, m∗d as a function of θ and Panel B does the same for the LR, Md∗ . The
dashed vertical line, which sits at b
θ = .4196 delimits the set of θs for which m∗d > 0 and the one for which
m∗d = 0.
49
Expected return of the risky asset at t=2 in the delayed trading equilibrium, R* (ω )
d
d
R*d(ωd)
4
3.5
m* >0
3
d
2.5
0.35
0.4
m*d=0
0.45
θ
Price of the risky asset at t=2 in the delayed trading equilibrium, P* (ω )
d
d
0.12
P*d(ωd)
0.11
0.1
0.09
0.08
0.35
*
md>0
0.4
θ
*
md=0
0.45
Figure 5. The top panel shows the expected return of the risky asset, Rd∗ (ω 20 , ω 2L ), as a function of θ at
date 2 in the delayed trading equilibrium. The bottom panel shows the price of the risky asset in states
∗
(ω 20 , ω 2L ), P2d
, as a function of θ at date 2 in the delayed trading equilibrium. The dashed vertical line
corresponds to b
θ = .4196.
50
*
π : The expected profit of the SRs in the delayed trading equilibrium
1.003
π*(θ)
1.002
*
*
md>0
md=0
1.001
1
0.35
0.4
0.45
θ
Π∗: The expected profit of the LRs in the delayed trading equilibrium
0.54
Π*(θ)
0.535
0.53
*
md>0
m*d=0
*
Πi
0.525
0.35
0.4
θ
0.45
Figure 6. Expected profits for the SR, π ∗ , (top panel) and the LR (bottom panel), Π∗ , as a function of θ in
the delayed trading equilibrium. The dashed vertical line corresponds to b
θ = .4196.
51
*
π : The expected profit of the SRs with and without commitment
1.004
π*(θ)
1.003
A
1.002
C
B
1.001
1
0.36
0.38
0.4
0.42
θ
0.44
0.46
0.48
Π∗: The expected profit of the LRs with and without commitment
0.54
Π*(θ)
0.535
C
B
A
0.53
0.525
0.36
0.38
0.4
0.42
θ
0.44
0.46
0.48
Figure 7. Expected profits for the SR, π ∗ , (top panel) and the LR (bottom panel), Π∗ , as a function of θ.
The first dashed vertical line corresponds to b
θ = .4196. The continuous line plots the expected profits
when the Pareto superior equilibrium is chosen. In regions A and B, the delayed trading equilibrium
exists and it is the Pareto superior equilibrium. In region C, which corresponds to θ ∈ (.4628, .4834],
the delayed trading equilibrium no longer exists as Pd∗ < δηρ and the sole equilibrium is the immediate
trading equilibrium. The dashed line corresponds to the expected profits when the SRs can commit to
liquidate assets in state ω 2L .
52
Π: The expected profit of the competitive and the monopolist LR
0.54
Π*(θ)
0.535
0.53
B
A
C
competitive
profits
monopoly
profits
0.525
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
θ
Prices in t=2 for the monopolist and the competitive case
0.48
0.1
A
B
competitive
prices
P∗d(ωd)
0.095
C
monopoly
prices
0.09
0.085
0.4
0.41
0.42
0.43
0.44
θ
0.45
0.46
0.47
0.48
Figure 8. Top panel: Expected profits of the monopolist (the thick line) and the competitive LR (the thin
∗
line) as a function of θ. Bottom panel: Prices in states (ω 20 , ω 2L ), P2d
, in the monopolist (the thick line)
and the competitive (the thin line) LR case.
53
π*d: The expected profit of the SRs
1.01
π*d(θ)
m*d>0
m*d=0
1.005
1
0.65
0.7
θ
0.75
0.8
Π∗: The expected profit of the LRs
Π*(θ)
0.68
0.675
Π*
0.67
d
*
Πx
0.665
0.66
0.65
0.7
θ
0.75
0.8
Figure 9. Top panel: Expected profits for the SR in the ex-ante contract when the outside value is the
expected profit associated with the delayed trading equilibrium. Bottom panel: Expected profits of the
LR in the ex-ante contract when the outside value of the SR is the expected profit associated with the
delayed trading equilibrium, Π∗x . Also included are the expected profit of the LR in the delayed trading
equilibrium, Π∗d , and in the immediate trading equilibrium, Π∗i .
54
Total cash position in the delayed trading equilibrium
m*d(θ)+M*d(θ)
0.8
0.6
0.4
0.2
0
0.65
*
M*x(θ)
md=0
0.7
0.75
θ
Cash position of the LRs in the ex−ante case
0.96
0.94
*
md>0
*
0.8
*
md>0
md=0
0.92
0.9
0.88
0.65
0.7
θ
0.75
0.8
Figure 10. Top panel: Total cash position, m∗d + Md∗ in the delayed trading equilibrium as a function of θ.
Bottom panel: Cash position of the LR in the ex-ante contract case as a function of θ when the outside
opportunity of the SR is the expected profit in the delayed trading equilibrium.
55